1,3

Also the number of irreducible set partitions of size n (see A055105) {1}; {1,2}; {1,2,3}, {1,23}; ...; and also the number of set partitions of n which do not have a proper subset of parts with a union equal to a subset {1,2,...,j} with j<n (atomic set partitions, see A087903) {1}; {12}; {13,2}, {123}; ...

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26.

M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

T. D. Noe, Table of n, a(n) for n=1..100

M. Klazar, Bell numbers, their relatives and algebraic differential equations

G.f.: 1-1/B(x) where B(x) = g.f. for A000110 the Bell numbers.

a(n) = Sum_{k = 1, ..., n-1}A087903(n, k). a(n+1) = Sum{k = 0..n} A086329(n, k) . a(n+2) = Sum_{k = 0..n} A086211(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 13 2004

G.f. x/(1-(x-x^2)/(1-x-(x-2x^2)/(1-2x-(x-3x^2)/...))) (a continued fraction). - Michael Somos Sep 22 2005

Hankel transform is A000142 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 21 2007

Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 26 2009: (Start)

G.f.: (of 1,1,2,6,.. ) 1/(1-x-x^2/(1-3x-2x^2/(1-4x-3x^2/(1-5x-4x^2/(1-6x-5x^2/(1-... (continued fraction);

G.f.: (of 1,2,6,.. ) 1/(1-2x-2x^2/(1-3x-3x^2/(1-4x-4x^2/(1-5x-5x^2/(1-... (continued fraction). (End)

m{1} = x1+x2+x3+..., so a(1) = 1

m{1,2} = x1 x2+x2 x1+x2 x3+x3 x2+x1 x3+..., m{12} = x1 x1+x2 x2+x3 x3+... where m{1} m{1} = m{1,2} + m{12}, so a(2)=2-1=1

m{1,2,3} = x1 x2 x3+x1 x2 x4+x1 x3 x4+..., m{12,3} = x1 x1 x2+x2 x2 x1+..., m{13,2} = x1 x2 x1+x2 x1 x2+..., m{1,23} = x1 x2 x2+x2 x1 x1+..., m{123}=x1 x1 x1+x2 x2 x2+... and there are 3 independent relations among these 5 elements m{12} m{1} = m{123} + m{12,3}, m{1} m{12} = m{123}+m{1,23}, m{1} m{1,1} = m{1,2,3}+m{12,3}+m{13,2} so a(3)=5-3=2

(PARI) a(n)=if(n<0, 0, polcoeff(1-1/serlaplace(exp(exp(x+x*O(x^n))-1)), n))

Row sums of A055105, A055106, A055107. Cf. A098742, A003319.

Row sums of A087903, A055105, A055106, A055107

Sequence in context: A014330 A124294 A124295 this_sequence A091768 A150274 A109317

Adjacent sequences: A074661 A074662 A074663 this_sequence A074665 A074666 A074667

nonn,easy,nice

Michael Somos

Edited by Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Sep 03 2005

0,3

Number of ordered partitions of n into n parts, allowing zeros (cf. A097070) is binomial(2*n-1,n) = a(n) = essentially A001700. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 15 2004

a(n) = A110556(n)*(-1)^n, central terms in triangle A110555. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005

Hankel transform is A000027; example: Det([1,1,3,10;1,3,10,35;3,10,35,126;10,35,126,462])=4 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 13 2007

a(n) is the number of functions f:[n]->[n] such that for all x,y in [n] if x<y then f(x)<=f(y). So 2*a(n)-n=A045992(n) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 02 2009]

Hankel transform of the aeration of this sequence is A000027 doubled: 1,1,2,2,3,3,... [From Paul Barry (pbarry(AT)wit.ie), Sep 26 2009]

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

a(n)=(0^n+C(2n, n))/2. - Paul Barry (pbarry(AT)wit.ie), May 21 2004

a(n) is the coefficient of x^n in 1/(1-x)^n and also the sum of the first n coefficients of 1/(1-x)^n. Given B(x) with the property that the coefficient of x^n in B(x)^n equals the sum of the first n coefficients of B(x)^n, then B(x)=B(0)/(1-x).

G.f.: 1/(2-C(x)) where C(x) is g.f. for Catalan numbers A000108.

G.f.: (1+1/sqrt(1-4x))/2. a(n)=binomial(2n-1,n).

a(n)=sum{k=0..n, binomial(2n, k)cos((n-k)*pi)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k))cos(k*pi/2)/2} (with interpolated zeros); a(n)=sum{k=0..floor(n/2), binomial(n, k)cos((n-2k)pi/2)} (with interpolated zeros); - Paul Barry (pbarry(AT)wit.ie), Nov 02 2004

a(n)=Sum_{k, 0<=k<=n}A094527(n,k)*(-1)^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 14 2007

The five rooted ordered trees with 3 edges have 10 leaves.

..x........................

..o..x.x..x......x.........

..o...o...o.x..x.o..x.x.x..

..r...r....r....r.....r....

seq(abs(binomial(-n, -2*n)), n=0..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007

(PARI) a(n)=sum(i=0, n, binomial(n+i-2, i))

(PARI) a(n)=if(n<0, 0, polcoeff((1+1/sqrt(1-4*x+x*O(x^n)))/2, n))

(PARI) a(n)=if(n<0, 0, polcoeff(1/(1-x+x*O(x^n))^n, n))

(PARI) a(n)=if(n<0, 0, binomial(2*n-1, n))

(PARI) {a(n)=if(n<1, n==0, polcoeff( subst((1-x)/(1-2*x), x, serreverse(x-x^2+x*O(x^n))), n))}

(Other) sage: [binomial(2*n+1, n+1) for n in xrange(-1, 24)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2009]

A001700(n)=a(n+1). a(n)=A024718(n)-A024718(n-1).

Sequence in context: A099908 A167403 A001700 this_sequence A110556 A072266 A085282

Adjacent sequences: A088215 A088216 A088217 this_sequence A088219 A088220 A088221

nonn

Michael Somos, Sep 24 2003

Essentially the same as A001700, which has much more information.

1,2

Dirichlet inverse of n.

Absolute values give n if n is square-free otherwise 0.

T. D. Noe, Table of n, a(n) for n=1..1000

Dirichlet g.f.: 1/zeta(s-1).

Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 17 2003

Conjectures: lim b->1+ Sum n=1..inf a(n)*b^(-n) = -12 and lim b->1- Sum n=1..inf a(n)*b^n = -12 (+ indicates that b decreases to 1, - indicates it increases to 1), both considering that zeta(-1) = -1/12 and calculations (more generally mu(n)*n^s is Abel summable to zeta(-s)). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Sep 26 2004

Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - Franklin T. Adams-Watters, Sep 11 2005.

(PARI) a(n)=n*moebius(n)

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1-p*X)[n])

Cf. A000027, A023900. Moebius transform of A023900.

Cf. A008683, A062004.

Sequence in context: A128214 A145105 A140700 this_sequence A049268 A004179 A122830

Adjacent sequences: A055612 A055613 A055614 this_sequence A055616 A055617 A055618

sign,easy,nice,mult

Michael Somos, Jun 04 2000

0,1

Index entries for two-way infinite sequences

G.f.: (2-2x+x^2)/((1-x)(1-x^2)). a(2n+1)=n. a(2n)=n+2. a(n+2)=a(n)+1. a(n)=-a(-3-n).

a(n) = floor(n/2) + 1 + (-1)^n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 27 2005

A112032(n)=2^a(n); A112033(n)=3*2^a(n); a(n)=A109613(n+2)-A052938(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 27 2005

a(n)=n-a(n-1) (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009]

For n=2, a(2)=2-2=0; n=3, a(3)=3-0=3; n=4, a(4)=4-3=1; n=5, a(5)=5-1=4 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 21 2009]

lst={}; a=1; Do[a=n-a; AppendTo[lst, a], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 14 2008]

(PARI) a(n)=n\2-2*(n%2)+2

(MAGMA) &cat[ [n+2, n]: n in [0..37] ]; [From Klaus Brockhaus, Nov 23 2009]

Cf. A030451.

Cf. A097065.

Cf. A152832 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 14 2008]

Sequence in context: A025636 A025637 A097065 this_sequence A008720 A008734 A053445

Adjacent sequences: A084961 A084962 A084963 this_sequence A084965 A084966 A084967

nonn

Michael Somos, Jun 15 2003

First part of definition adjusted to match offset by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 23 2009

0,12

M. Somos, Sequences used for indexing triangular or square arrays

0; 0; 0; 0,1; 0,1; 0,1; 0,1,2; 0,1,2; 0,1,2; 0,1,2,3; 0,1,2,3; ...

(PARI) a(n)=local(m); m=floor(sqrt(6*n+6)-3/2)\3+1; (n-3*binomial(m, 2))%m

Sequence in context: A025858 A025684 A025678 this_sequence A025855 A097203 A025850

Adjacent sequences: A073186 A073187 A073188 this_sequence A073190 A073191 A073192

nonn

Michael Somos, Jul 19 2002

0,2

Apart from signs, identical to A053122.

Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004

T(n,k)=number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2)=10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005

T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72 (2006), 51-62. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]

T. D. Noe, Rows n=0..50 of triangle, flattened

G.f.: x*y/(1-(2+y)*x+x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.

If indexing begins at 0 we have: T(n, k) = (n+k+1)!/((n-k)!*(2k+1))!. T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n<k. T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n, k) = 0 if k<0, T(0, 0)=1 and T(0, k) = 0 for k>0. G.f. for the column k : Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2). Row sums : Sum_{k>=0} T(n, k) = A001906(n+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004

Diagonal sums are A000079(n)=sum{k=0..floor(n/2), binomial(n+k+1, n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004

Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), Oct 22 2006

T(0,0)= 1, T(n,k)=0 if k<0 or if k>n, T(n,k) = T(n-1,k-1)+2*T(n-1,k)-T(n-2,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 26 2010]

Triangle begins:

.........................1

........................2,.1

......................3,.4,.1

....................4,.10,.6,.1

..................5,.20,.21,.8,.1

................6,.35,.56,.36,.10,.1

.............7,.56,.126,.120,.55,.12,.1

..........8,.84,.252,.330,.220,.78,.14,.1

for n from 1 to 11 do seq(binomial(n+k-1, 2*k-1), k=1..n) od; # yields sequence in triangular form (Deutsch)

(PARI) T(n, k)=if(n<0, 0, binomial(n+k-1, 2*k-1))

(PARI) {T(n, k)=polcoeff( polcoeff( x*y/(1-(2+y)*x+x^2) +x*O(x^n), n), k)}

This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.

The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.

Cf. A053123, A049310. Row sums give A001906.

With signs: A053122.

Cf. A119900 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]

Sequence in context: A137614 A143326 A053122 this_sequence A104711 A133112 A159856

Adjacent sequences: A078809 A078810 A078811 this_sequence A078813 A078814 A078815

easy,nice,nonn,tabl,new

Michael Somos, Dec 05, 2002

Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 28 2008

0,13

Computing each term modulo 2 also gives A047999, i.e. A051159[n] mod 2 = A007318[n] mod 2 for all n. (The triangle is paritywise isomorphic to Pascal's Triangle) - Antti Karttunen

5th row/column gives entries of A000217 (triangular numbers C(n+1,2)) repeated twice and every other entry in 6th row/column form A000217. 7th row/column gives entries of A000292 (Tetrahedral (or pyramidal) nos: C(n+3,3)) repeated twice and every other entry in 8th row/column form A000292. 9th row/column gives entries of A000332 (binomial coefficients binomial(n,4)) repeated twice and every other entry in 10th row/column form A000332. 11th row/column gives entries of A000389 (binomial coefficients C(n,5)) repeated twice and every other entry in 12th row/column form A000389. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 21 2004

If Sum_{k=0..n}A(k)*T(n,k)=B(n), the sequence B is the S-D transform of the sequence A . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2006

Number of n-bead black-white reversible strings with k black beads; also binary grids; string is palindromic. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008

Row sums give A016116(n+2) - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008

Coefficients of expansion of (x+y)^n where x and y anticommute (yx = -xy), that is, q-binomial coefficients when q = -1. - Michael Somos Feb 16 2009

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)

The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=0.

Row sums are:

{1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64,...} (End)

S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

M. E. Horn, The Didactical Relevance of the Pauli Pascal Triangle [From Michael Somos]

T(n, k)=T(n-1, k-1)+T(n-1, k) if n odd or k even, else 0. T(0, 0)=1.

T(n, k)=T(n-2, k-2)+T(n-2, k). T(0, 0)=T(1, 0)=T(1, 1)=1.

Square array made by setting first row/column to 1's (A(i, 0) = A(0, j) = 1); A(1, 1) = 0; A(1, j) = A(1, j-2); A(i, 1) = A(i-2, 1); other entries A(i, j) = A(i-2, j) + A(i, j-2). - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 21 2004

Sum_{k=0..n}k*T(n,k)=A093968(n); A093968 = S-D transform of A001477 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 02 2006

Equals 2*A034851 - A007318, - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 31 2007. [Corrected by Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008]

w-0:\q p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009]

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)

{1},

{1, 1},

{1, 0, 1},

{1, 1, 1, 1},

{1, 0, 2, 0, 1},

{1, 1, 2, 2, 1, 1},

{1, 0, 3, 0, 3, 0, 1},

{1, 1, 3, 3, 3, 3, 1, 1},

{1, 0, 4, 0, 6, 0, 4, 0, 1},

{1, 1, 4, 4, 6, 6, 4, 4, 1, 1},

{1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1},

{1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1} (End)

Contribution from Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009: (Start)

Clear[p, n, x, a]

w = 0;

p[x, 1] := 1;

p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]]

a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]

Flatten[a] (End)

(PARI) {T(n, k) = binomial(n%2, k%2) * binomial(n\2, k\2)} [From Michael Somos]

Cf. A007318. A051160(n, k)=(-1)^[ k/2 ]*A051159(n, k).

Cf. A016116, A034851.

Cf. A169623 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Dec 04 2009]

Sequence in context: A035196 A158020 A051160 this_sequence A035697 A135549 A124737

Adjacent sequences: A051156 A051157 A051158 this_sequence A051160 A051161 A051162

nonn,tabl,easy,nice

Michael Somos

0,5

This is an example of a sequence of Lehmer numbers. In this case, the two parameters, alpha and beta, are (1 +- i Sqrt(7))/2. Bilu, Hanrot, Voutier and Mignotte show that all terms of a Lehmer sequence a(n) have a primitive factor for n > 30. Note that for this sequence, a(30) = 24475 = 5*5*11*89 has no primitive factors. - T. D. Noe (noe(AT)sspectra.com), Oct 29 2003

Row sums of Riordan array (1/(1+2x^2),x/(1+2x^2)). - Paul Barry (pbarry(AT)wit.ie), Sep 10 2005

Y. Bilu, G. Hanrot, P. M. Voutier and M. Mignotte, Existence of primitive divisors of Lucas and Lehmer numbers

Eric Weisstein's World of Mathematics, Lehmer Number

G.f.; x/(1-x+2x^2). a(n)=a(n-1)-2*a(n-2).

a(n+1)=sum{k=0..n, C((n+k)/2, k)*(-2)^((n-k)/2)*(1+(-1)^(n-k))/2}; a(n+1)=sum{k=0..floor(n/2), C(n-k, k)(-2)^k}; - Paul Barry (pbarry(AT)wit.ie), Sep 10 2005

a(n+1)=Sum_{k, 0<=k<=n} A109466(n,k)*2^(n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 26 2008]

a:= n-> (Matrix([[1, 1], [ -2, 0]])^n)[1, 2]: seq (a(n), n=0..45); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 03 2008]

(PARI) a(n)=if(n<0, 0, imag(quadgen(-7)^n))

(Other) sage: [lucas_number1(n, 1, +2) for n in xrange(0, 46)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]

A001607(n)=-(-1)^n*a(n).

Sequence in context: A001607 A167433 A077020 this_sequence A159285 A021080 A049764

Adjacent sequences: A107917 A107918 A107919 this_sequence A107921 A107922 A107923

sign

Michael Somos, May 28 2005

0,1

Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized pentagonal numbers: a(A001318(n))=1, a(A118300(n))=0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 22 2006

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 81, Article 331.

T. D. Noe, Table of n, a(n) for n=0..1001

Index entries for characteristic functions

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal numbers, A000326].

a(n)=b(24n+1) where b(n) is multiplicative and b(2^e)=b(3^e)=0^e, b(p^e)=(1+(-1)^e)/2 if p>3. - Michael Somos Jun 06 2005

Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].

Expansion of phi(-q^3) / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions. - Michael Somos Sep 14 2007

Expansion of psi(q) - q * psi(q^9) in powers of q^3 where psi() is a Ramanujan theta function. - Michael Somos Sep 14 2007

Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable theta function.

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 pi i t) and g(t) is g.f. for A089810.

Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q.

G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - Michael Somos Jan 26 2008

q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...

(PARI) a(n)=if(n<0, 0, abs(polcoeff(eta(x+x*O(x^n)), n)))

(PARI) a(n)=issquare(24*n+1) /* Michael Somos Apr 13 2005 */

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^3+A)^2/eta(x+A)/eta(x^6+A), n))}

|A010815(n)| = a(n). A089806(2n) = a(n). A033683(24n+1) = a(n).

Sequence in context: A115513 A133080 A010815 this_sequence A121373 A133985 A143062

Adjacent sequences: A080992 A080993 A080994 this_sequence A080996 A080997 A080998

nonn,easy

Michael Somos, Feb 27, 2003

0,4

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 345 Entry 1(i).

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)

G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(6k-3))^3.

Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^2-v+2uv^2.

Given g.f. A(x), then B(x)=x*A(x^3) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(v^3-v^2+v)-u^3(1+2v+4v^2).

Expansion of q^(-1/3)eta(q)eta(q^6)^3/(eta(q^2)eta(q^3)^3) in powers of q.

Euler transform of period 6 sequence [ -1,0,2,0,-1,0,...].

Expansion of chi(-q)/chi(-q^3)^3 where chi() (g.f. A000700) is a Ramanujan theta function.

G.f.: 1/(1+ (x+x^2)/(1+ (x^2+x^4)/(1 +(x^3+x^6)/...))).

Expansion of q^(-1/3)c(q^2)/c(q) where c() is a cubic AGM analog function. - Michael Somos Oct 04 2006

q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...

(PARI) {a(n)=local(A, m); if(n<0, 0, A=1+O(x); m=1; while(m<=n, m*=2; A=subst(A, x, x^2); A=sqrt(A+(x*A^2)^2)-x*A^2); polcoeff(A, n))}

(PARI) {a(n)=if(n<0, 0, polcoeff(prod(k=0, (n-1)\2, (1-x^(2*k+1))^if(k%3==1, -2, 1), 1+x*O(x^n)), n))}

A062242(2*n + 1) = a(n). A128111(n) = (-1)^n * a(n). Convolution inverse of A062242.

Sequence in context: A129862 A138189 A110090 this_sequence A128111 A107356 A124725

Adjacent sequences: A092845 A092846 A092847 this_sequence A092849 A092850 A092851

sign

Michael Somos, Mar 07 2004