0,3

T(n,k) + T(n,n-k) = A014105(n);

row sums give A059270; Sum(T(n,k): 0<=k<n) = A000578(n);

central terms give A007742; T(2*n+1,n) = A016754(n);

T(n,0) = A000217(n);

T(n,1) = A000096(n) for n>0;

T(n,2) = A055998(n) for n>1;

T(n,3) = A055999(n) for n>2;

T(n,4) = A056000(n) for n>3;

T(n,5) = A056115(n) for n>4;

T(n,6) = A056119(n) for n>5;

T(n,7) = A056121(n) for n>6;

T(n,8) = A056126(n) for n>7;

T(n,10) = A101859(n-1) for n>9;

T(n,n-3) = A095794(n-1) for n>2;

T(n,n-2) = A045943(n-1) for n>1;

T(n,n-1) = A000326(n) for n>0;

T(n,n) = A005449(n).

Cf. A110449.

Sequence in context: A081622 A064143 A115274 this_sequence A122637 A076229 A160102

Adjacent sequences: A126887 A126888 A126889 this_sequence A126891 A126892 A126893

nonn,tabl

Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

0,2

a(n)=A000096 + 7 * A001477, a(n)=A056121 + A001477 and a(n)=A051942 - A001477 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 01 2006

a(n) = A126890(n,8) for n>7. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

G.f.(x)=x(9-8x)/(1-x)^3.

a(n)=C(n,2)-8*n,n>=17 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2006

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = -f(n,n-1,9), for n>=1. [From Milan R. Janjic (agnus(AT)blic.net), Dec 20 2008]

a(n)=n+a(n-1)+7 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 19 2009]

For n=2, a(2)=2+0+7=9; n=3, a(3)=3+9+7=19; n=4, a(4)=4+19+7=30 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 19 2009]

a:=n->sum(floor(k+2*n/(k+n)), k=8..n): seq(a(n), n=7..57); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 01 2006

[seq(binomial(n, 2)-8*n, n=17..67)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2006

a:=n->sum(n, j=18..n): seq(a(n)/2, n=17..67); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 17 2008

a:=n->sum(denom (k/(k+3)), k=6..n): seq(a(n), n=5..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2008

with(finance):seq(add(cashflows([2, k, 6], 0 ), k=1..n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008

i=-8; s=0; lst={}; Do[s+=n+i; If[s>=0, AppendTo[lst, s]], {n, 0, 6!, 1}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 29 2008]

Cf. A056121.

Cf. A000096, A056121, A051942, A056000, A001477.

Cf. A001477, A098849, A120071, A132760, A132761, A132765.

Sequence in context: A043525 A031499 A017377 this_sequence A051811 A034056 A157034

Adjacent sequences: A056123 A056124 A056125 this_sequence A056127 A056128 A056129

easy,nonn

Barry E. Williams, Jul 07 2000

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000

0,6

Mirror image of triangle A133336 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2008]

Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deleham's operator defined in A084938.

For k>0, T(n, k) = binomial(n+k-1, n)*binomial(n+2k, k)/(n+k+1); T(0, 0) = 1 and T(n, 0) = 0 if n>0.

Sum_{k>=0} T(n, k)*2^k = A107841(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 26 2005

Sum_{ k>=0} T(n-k, k) = A005043(n) . - Philippe DELEHAM, May 30 2005

T(n, k) = A108263(n+k, k) . - Philippe DELEHAM, May 30 2005

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2007

Sum_{k, 0<=k<=n}T(n,k)*5^k*(-2)^(n-k) = A152601(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2008]

Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*3^(n-k) = A154825(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 17 2009]

1; 0, 1; 0, 1, 2; 0, 1, 5, 5; 0, 1, 9, 21, 14; ...

Diagonals : A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281

The diagonals (except for A000007) are also the diagonals of A033282.

Row sums : A001003 (Schroeder numbers)

Cf. A033282, A084938.

Sequence in context: A004483 A085650 A109450 this_sequence A085838 A094456 A010028

Adjacent sequences: A086807 A086808 A086809 this_sequence A086811 A086812 A086813

easy,nonn,tabl

DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 05 2003

0,5

The divergent series g(x,m) = sum((-1)^(k+1)*k^m*k!*x^k, k= 1..infinity), m=>-1, are related to the higher order exponential integrals E(x,m,n=1), see A163931.

Hardy, see the link below, describes how Euler came to the rather surprising conclusion that g(x,-1) = exp(1/x)*Ei(1,1/x) with Ei(1,x) = E(x,m=1,n=1). From this result it follows inmediately that g(x,0) = 1 - g(x,-1). Following in Euler's footsteps we discovered that g(x,m) = (-1)^(m) * (M(x,m)*x - ST(x,m)* Ei(1,1/x) * exp(1/x))/x^(m+1), m =>-1.

So g(x=1,m) = (-1)^m*(A040027(m) - A000110 (m+1)*A073003), with A040027(m = -1) = 0. We observe that A073003 = - exp(1)*Ei(-1) is Gompertz's constant, A000110 are the Bell numbers and A040027 was published a few years ago by Gould.

The polynomial coefficients of the M(x,m) = sum(a(m,k) * x^k, k = 0..m), for m =>0 lead to the triangle given above. We point out that M(x,m=-1) = 0.

The polynomial coefficients of the ST(x,m) = sum(S2(m+1, k) * x^k, k = 0..m+1), m =>-1, lead to the Stirling numbers of the second kind, see A106800.

The formulae that generate the coefficients of the left hand columns lead to the Minkowski numbers A053657. We have a closer look at them in A163972.

The right hand columns have simple generating functions, see the formulae. We used them in the first Maple program to generate the triangle coefficients (n >= 0 and 0 <= k <= n). The second Maple program calculates the values of g(x,m) for m=>-1, at x=1.

G.H. Hardy, Divergent Series, Oxford University Press, 1949. pp. 26-29 and pp. 7-8.

The generating functions of the right hand columns are Gf(p) = 1/((1-(p-1)*x)^2*product((1-k*x), k=1..p-2)); Gf(1) = 1. For the first right hand column p=1, for the second p=2, etc..

The first few triangle rows are:

[1]

[1, 0]

[1, 2, 0]

[1, 5, 3, 0]

[1, 9, 17, 4, 0]

[1, 14, 52, 49, 5, 0]

The first few M(x,m) are:

M(x,m=0) = 1

M(x,m=1) = 1 + 0*x

M(x,m=2) = 1 + 2*x + 0*x^2

M(x,m=3) = 1 + 5*x + 3*x^2 + 0*x^3

The first few ST(x,m) are:

ST(x,m=-1) = 1

ST(x,m=0) = 1 + 0*x

ST(x,m=1) = 1 + 1*x + 0*x^2

ST(x,m=2) = 1 + 3*x + x^2 + 0*x^3

ST(x,m=3) = 1 + 6*x + 7*x^2 + x^3 + 0*x^4

The first few g(x,m) are:

g(x,-1) = (-1)*(- (1)*Ei(1,1/x)*exp(1/x))/x^0

g(x,0) = (1)*((1)*x - (1)*Ei(1,1/x)*exp(1/x))/x^1

g(x,1) = (-1)*((1)*x - (1+ x)*Ei(1,1/x)*exp(1/x))/x^2

g(x,2) = (1)*((1+2*x)*x - (1+3*x+x^2)*Ei(1,1/x)*exp(1/x))/x^3

g(x,3) = (-1)*((1+5*x+3*x^2)*x - (1+6*x+7*x^2+x^3)*Ei(1,1/x)*exp(1/x))/x^4

restart; nmax:=11; imax:=nmax: for p from 1 to imax do Gf(p):=convert(series(1/((1-(p-1)*x)^2*product((1-k*x), k=1..p-2)), x, imax+1-p), polynom); for q from 0 to imax-p do a(p+q-1, q):=coeff(Gf(p), x, q) od: od: T:=0: for n from 0 to nmax-1 do for k from 0 to n do a(T):=a(n, k); T:=T+1; od: od: seq(a(n), n=0..T-1);

restart; nmax:=11; A040027(-1):=0: A040027(0):=1: for n from 1 to nmax do A040027(n) := sum(binomial(n, k-1)*A040027(n-k), k = 1..n) od: A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i)*A000110(n-1-i), i=0..n-1); fi; end: A073003 := - exp(1) * Ei(-1): for n from -1 to nmax do g(1, n):= (-1)^n*(A040027(n)-A000110(n+1)*A073003) od;

The row sums equal A040027 (Gould).

A000007, A000027, A000337, A163941 and A163942 are the first five right hand columns.

A000012, A000096, A163943 and A163944 are the first four left hand columns.

Cf. A163931, A163972, A106800 (Stirling2), A000110 (Bell), A073003 (Gompertz), A053657 (Minkowski).

Sequence in context: A125183 A092583 A079134 this_sequence A112340 A037186 A004483

Adjacent sequences: A163937 A163938 A163939 this_sequence A163941 A163942 A163943

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009

3,3

T(n+3,k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example x^2+5*x+5=y^2+3*y+1. - Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003

Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).

Number of k dimensional 'faces' of the n dimensional associahedron (see Simion, p. 168). - Mitch Harris (maharri(AT)gmail.com), Jan 16 2007

Mirror image of triangle A126216 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 19 2007

For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]

Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.

A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)

B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.

S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

G. Kreweras, Sur les partitions..., Discrete Math. 1 (1972), 333-350.

R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149-180.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. [From Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008]

F. Chapoton, Enumerative properties of generalized associahedra

P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529.

S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

G.f. G=G(t, z) satisfies (1+t)G^2-z(1-z-2tz)G+tz^4=0.

T(n, k)=binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <=k <=n-3.

Contribution from Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008: (Start)

Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.

1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)]

= t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...

b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)}

= (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...

2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)]

= (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...

b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)]

= (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...

c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)]

= t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...

x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437)

to generate A033282 and show that A133437 is a refinement of A033282,

i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.

y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264)

to generate A033282 and show that A134264 is a refinement of A001263, i.e.,

a refinement of the h-polynomials of the associahedra.

f1[x,t](t+1) gives a generator for A088617.

f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].

f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].

The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)

G.f.: 1/(1-xy-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 06 2009]

1; 1,2; 1,5,5; 1,9,21,14; 1,14,56,84,42;

Diagonals : A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281 Row sums : A001003 (Schroeder numbers, first term omitted) . See A086810 for another version.

A007160 is a diagonal. Cf. A001263.

With leading zero: A086810.

Cf. A019538 'faces' of the permutohedron.

Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image). [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]

Sequence in context: A145882 A111785 A021468 this_sequence A126350 A079502 A126124

Adjacent sequences: A033279 A033280 A033281 this_sequence A033283 A033284 A033285

nonn,tabl,easy

N. J. A. Sloane (njas(AT)research.att.com).

Added a missing factor of 2 for expansions of f1 and f2 Tom Copeland (tcjpn(AT)msn.com), Apr 12 2009

0,2

a(n)=A000096 + 3 * A001477, a(n)=A055999 + A001477 and a(n)=A056115 - A001477 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 01 2006

a(n) = A126890(n,4) for n>3. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

G.f.(x)=x(5-4x)/(1-x)^3.

a(n)=C(n,2)-4*n,n>=9 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

Equals A028569/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 12 2007

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = -f(n,n-1,5), for n>=1. [From Milan R. Janjic (agnus(AT)blic.net), Dec 20 2008]

a(n)=n+a(n-1)+3 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 19 2009]

For n=2, a(2)=2+0+3=5; n=3, a(3)=3+5+3=11; n=4, a(4)=4+11+3=18 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 19 2009]

a:=n->sum(floor(k+2*n/(k+n)), k=4..n): seq(a(n), n=3..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 01 2006

[seq(binomial(n, 2)-4*n, n=9..59)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

a:=n->sum(n/2, j=10..n): seq(a(n), n=9..51); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 12 2007

seq(sum(k, k=5..n), n=4..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

a:=n->sum(numer (k/(k+3)), k=5..n): seq(a(n), n=4..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2008

with(finance):seq(add(cashflows([k, k, 8], 0 ), k=1..n)/2, n=0..45); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 22 2008]

lst={}; Do[AppendTo[lst, n*(n+9)/2], {n, 0, 5!}]; lst ...and/or... s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 4, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]

Equals A000217(n+4)-10. Cf. A000096, A055998 and A055999.

Column m=2 of (1, 5)-Pascal triangle A096940.

Cf. A000096, A055998, A056000, A001477.

Sequence in context: A166039 A145005 A004083 this_sequence A080566 A094684 A140697

Adjacent sequences: A055997 A055998 A055999 this_sequence A056001 A056002 A056003

easy,nonn

Barry E. Williams, Jun 16 2000

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 04 2000

0,2

a(n) is the smallest non-negative integer not yet in the sequence such that n + a(n) is one less than a square. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Apr 06 2009]

Harry J. Smith, Table of n, a(n) for n=0,...,1000

Index entries for sequences that are permutations of the natural numbers

a(n) =[sqrt(2n+1)-1/2]*[sqrt(2n+1)+3/2]-n =A005563(A003056(n))-n

(PARI) { default(realprecision, 100); for (n=0, 1000, f=floor(sqrt(2*n + 1) - 1/2); write("b061579.txt", n, " ", f*(f + 2) - n) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 25 2009]

Fixed points are A046092. Each reversal involves the numbers from A000217 through to A000096.

A self-inverse permutation of the nonnegative numbers.

Cf. A038722. Transpose of A001477.

Sequence in context: A113350 A164678 A164679 this_sequence A094064 A159930 A058344

Adjacent sequences: A061576 A061577 A061578 this_sequence A061580 A061581 A061582

nonn,tabl

Henry Bottomley (se16(AT)btinternet.com), May 21 2001

0,2

a(n) is also the dimension of the irreducible representation of the Lie algebra sl(3) with the highest weight 2*L_1+n*(L_1+L_2) [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Jan 04 2010]

W. FULTON, J. HARRIS. Representation theory: a first course. (1991). page 224, Exercise 15.19. [From Leonid Bedratyuk (leonid.uk(AT)gmail.com), Jan 04 2010]

a(n) = A000096(n)*3 = (3*n^2 + 9*n)/2 = n(3n+9)/2.

a(n)=3*n+a(n-1) (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 28 2009]

For n=2, a(2)=3*2+0=6; n=3, a(3)=3*3+6=15; n=4, a(4)=3*4+15=27 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 28 2009]

with(finance):seq(add(cashflows([2, k, n], 0 ), k=2..n), n=1..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 22 2008

s=0; lst={}; Do[s+=n+1; s+=n+2; s+=n+3; AppendTo[lst, s], {n, 0, 4!, 1}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 30 2008]

lst={}; Do[AppendTo[lst, 3*n*(n+3)/2], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 06 2008]

Table[Sum[i + n - 3, {i, 0, n}], {n, 2, 45}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]

Cf. A000096, A000326, A005449, A045943, A115067, A140090, A059845, A140672, A140673, A140674, A140675.

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Sequence in context: A022601 A112150 A072257 this_sequence A165454 A063525 A161777

Adjacent sequences: A140088 A140089 A140090 this_sequence A140092 A140093 A140094

easy,nonn

Omar E. Pol (info(AT)polprimos.com), May 22 2008

0,2

Sum of n terms of the arithmetic progression with first term 1 and common difference n-1. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 04 2005

a(n) = sum of row (n+1)-th row terms of triangle A144693. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 19 2008]

Harry J. Smith, Table of n, a(n) for n=0,...,1000

a(n)=(n+1)(n^2+2)/2

a(n)=sum{k=0..n, sum{j=0..n, (k-(k-1)*C(0, j-k)}}; a(n)=A006002(n)-A000096(n-2). - Paul Barry (pbarry(AT)wit.ie), Nov 18 2005

G.f.: (1-x+3x^2)/(1-x)^4. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009]

(PARI) { for (n=0, 1000, write("b064808.txt", n, " ", (n + 1)*(n^2 + 2)/2) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Sep 26 2009]

Main diagonal of A057145.

A144693 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 19 2008]

Sequence in context: A063586 A131477 A002128 this_sequence A001937 A086817 A000715

Adjacent sequences: A064805 A064806 A064807 this_sequence A064809 A064810 A064811

easy,nonn

Floor van Lamoen (fvlamoen(AT)hotmail.com), Oct 22 2001

0,2

a(n) = A126890(n,3) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 30 2006

If X is an n-set and Y a fixed (n-4)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Aug 15 2007

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

Milan Janjic, Two Enumerative Functions

G.f.(x)=x(4-3x)/(1-x)^3.

a(n)=C(n,2)-3*n,n>=7 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

Equals A028563/2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 12 2007

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = -f(n,n-1,4), for n>=1. [From Milan R. Janjic (agnus(AT)blic.net), Dec 20 2008]

a(n)=n+a(n-1)+2 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 18 2009]

For n=2, a(2)=2+0+2=4; n=3, a(3)=3+4+2=9; a(4)=4+9+2=15 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 18 2009]

a:=n->sum(floor(k+2*n/(k+n)), k=3..n): seq(a(n), n=2..53); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 01 2006

[seq(binomial(n, 2)-3*n, n=7..58)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

a:=n->sum(n/2, j=8..n): seq(a(n), n=7..58); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 12 2007

seq(sum(k-1, k=5..n), n=4..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 28 2008

a:=n->sum(numer (k/(k+3)), k=4..n): seq(a(n), n=3..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 31 2008

with (combinat):seq((fibonacci(3, n)+n-13)/2, n=3..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008

lst={}; Do[AppendTo[lst, n*(n+7)/2], {n, 0, 5!}]; lst ...and/or... s=0; lst={s}; Do[s+=n+1; AppendTo[lst, s], {n, 3, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]

Equals A000217(n+3)-6. Cf. A000096, A055998, A074171.

Third column (m=2) of (1, 4)-Pascal triangle A095666.

Cf. A000096, A055998, A056000, A001477.

Cf. A002522.

Sequence in context: A073046 A066495 A134227 this_sequence A022945 A022443 A079423

Adjacent sequences: A055996 A055997 A055998 this_sequence A056000 A056001 A056002

easy,nonn

Barry E. Williams, Jun 16 2000

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 04 2000