Search: id:A039599
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%I A039599
%S A039599 1,1,1,2,3,1,5,9,5,1,14,28,20,7,1,42,90,75,35,9,1,132,297,275,154,
%T A039599 54,11,1,429,1001,1001,637,273,77,13,1,1430,3432,3640,2548,1260,
%U A039599 440,104,15,1,4862,11934,13260,9996,5508,2244,663,135,17,1
%N A039599 Triangle formed from even-numbered columns of triangle of expansions
of powers of x in terms of Chebyshev polynomials U_n (x).
%C A039599 T(n,k) is the number of lattice paths from (0,0) to (n,n) with steps
E = (1,0) and N = (0,1) which touch but do not cross the line x -
y = k and only situated above this line; example : T(3,2) = 5 because
we have EENNNE, EENNEN, EENENN, ENEENN, NEEENN. - Philippe DELEHAM,
May 23 2005
%C A039599 The matrix inverse of this triangle is the triangular matrix T(n,k) =
(-1)^(n+k)* A085478(n,k). - Philippe DELEHAM, May 26 2005
%C A039599 Essentially the same as A050155 except with a leading diagonal A000108
(Catalan numbers) 1, 1, 2, 5, 14, 42, 132, 429, . . . - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), May 31 2005
%C A039599 Number of Grand Dyck paths of semilength n and having k downward returns
to the x-axis. (A Grand Dyck path of semilength n is a path in the
half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting
of steps u=(1,1) and d=(1,-1)). Example: T(3,2)=5 because we have
u(d)uud(d),uud(d)u(d),u(d)u(d)du,u(d)duu(d) and duu(d)u(d) (the downward
returns to the x-axis are shown between parentheses). - Emeric Deutsch
(deutsch(AT)duke.poly.edu), May 06 2006
%C A039599 Riordan array (c(x),x*c(x)^2) where c(x) is the g.f. of A000108 ; inverse
array is (1/(1+x),x/(1+x)^2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Feb 12 2007
%C A039599 The triangle may also be generated from M^n*[1,0,0,0,0,0,0,0,...], where
M is the infinite tridiagonal matrix with all 1's in the super and
subdiagonals and [1,2,2,2,2,2,2,...] in the main diagonal . - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Feb 26 2007
%C A039599 Inverse binomial matrix applied to A124733 . Binomial matrix applied
to A089942 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 26
2007
%C A039599 Number of standard tableaux of shape (n+k,n-k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 22 2007
%C A039599 Comment from Phipppe DELEHAM, Mar 30 2007: This triangle belongs to the
family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>
n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,
k+1) for k>=1 . Other triangles arise by choosing different values
for (x,y):
%C A039599 (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970;
(1,0) -> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877;
(1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598;
(2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954;
(3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791;
(4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906.
%C A039599 The table U(n,k)=Sum_{j, 0<=j<=n}T(n,j)*k^j is given in A098474 . - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Mar 29 2007
%C A039599 Sequence read mod 2 gives A127872 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Apr 12 2007
%C A039599 Number of 2n step walks from (0,0) to (2n,2k)and consisting of step u=(1,
1) and d=(1,-1) and the path stays in the nonnegative quadrant .
Example :T(3,0)=5 because we have uuuddd, uududd, ududud, uduudd,
uuddud ; T(3,1)=9 because we have uuuudd, uuuddu, uuudud, ududuu,
uuduud, uduudu, uudduu, uduuud, uududu ; T(3,2)=5 because we have
uuuuud, uuuudu, uuuduu, uuduuu, uduuuu ; T(3,3)=1 because we have
uuuuuu . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007,
Apr 17 2007, Apr 18 2007
%C A039599 Triangular matrix, read by rows, equal to the matrix inverse of triangle
A129818 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 19 2007
%C A039599 Let Sum_{n>=0}a(n)*x^n = (1+x)/(1-mx+x^2)= o.g.f. of A_m, then Sum_{k,
0<=k<=n}T(n,k)*a(k)= (m+2)^n. Related expansions of A_m are : A099493,
A033999, A057078, A057077, A057079, A005408, A002878, A001834, A030221,
A002315, A033890, A057080, A057081, A054320, A097783, A077416, A126866,
A028230, A161591, for m= -3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,
14,15 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 16 2009]
%D A039599 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 796 W.-J. Woan, L. Shapiro and D. G. Rogers, The
Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly,
104 (1997), 926-931.
%D A039599 Paul Barry, A Catalan Transform and Related Transformations on Integer
Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A039599 T. D. Noe, Rows n=0..50 of triangle, flattened
%H A039599 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%F A039599 Row n: C(2n-1, n-k)-C(2n-1, n-k-2).
%F A039599 Triangle T(n, k) read by rows; given by A000012 DELTA A000007, where
DELTA is Deleham's operator defined in A084938.
%F A039599 T(n, k) = C(2*n, n-k)*(2*k+1)/(n+k+1). Sum(k>=0; T(n, k)*T(m, k) = A000108(n+m));
A000108: numbers of Catalan. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Dec 22 2003
%F A039599 T(n, 0) = A000108(n); T(n, k) = 0 if k>n; for k>0, T(n, k) = Sum_{j=1..n)
T(n-j, k-1)*A000108(j). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Feb 03 2004
%F A039599 T(n, k) = A009766(n+k, n-k) = A033184(n+k+1, 2k+1). - DELEHAM Philippe
(kolotoko(AT)wanadoo.fr), Feb 03 2004
%F A039599 G.f. for column k: Sum_{n>=0} T(n, k)*x^n = x^k*C(x)^(2*k+1) where C(x)
= Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108.
- DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 03 2004
%F A039599 T(0, 0) = 1, T(n, k) = 0 if n<0 or n=1, T(n, k) = T(n-1, k-1) + 2*T(n-1, k) + T(n-1, k+1).
- DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004
%F A039599 a(n) + a(n+1) = 1 + A000108(m+1) if n = m*(m+3)/2; a(n) + a(n+1) = A039598(n)
otherwise. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 18 2004
%F A039599 T(n, k) = A050165(n, n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Feb 27 2004
%F A039599 Sum_{j>=0} T(n-k, j)*A039598(k, j) = A028364(n, k). - DELEHAM Philippe
(kolotoko(AT)wanadoo.fr), Mar 04 2004
%F A039599 Matrix inverse of the triangle T(n, k) = (-1)^(n+k)*binomial(n+k, 2*k)
= (-1)^(n+k)*A085478(n, k). - Philippe Deleham (kolotoko(AT)wanadoo.fr)
%F A039599 Sum_{k, 0<=k<=n} T(n, k)*x^k = A000108(n), A000984(n), A007854(n), A076035(n),
A076036(n) for x = 0, 1, 2, 3, 4 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 19 2005
%F A039599 Sum_{k, 0<=k<=n} (2*k+1)*T(n, k) = 4^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 22 2005
%F A039599 T(n, k)*(-2)^(n-k) = A114193(n, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 17 2005
%F A039599 Sum_{k>=h}T(n,k)=binomial(2n,n-h). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Apr 30 2006
%F A039599 T(n,k)=(2k+1)*binomial(2n,n-k)/(n+k+1). G.f.=G(t,z)=1/[1-(1+t)zC], where
C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
May 06 2006
%F A039599 Sum_{k, 0<=k<=n} T(n,k)*5^k = A127628(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jan 22 2007
%F A039599 Sum_{k, 0<=k<=n} T(n,k)*7^k = A115970(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jan 26 2007
%F A039599 T(n,k)=Sum_{j, 0<=j<=n-k}A106566(n+k,2*k+j) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Feb 12 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*6^k = A126694(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Feb 16 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A000108(k)=A007852(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 22 2007
%F A039599 Sum_{k, 0<=k<=[n/2]}T(n-k,k)=A000958(n+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 22 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*(-1)^k=A000007(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 22 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*(-2)^k = (-1)^n*A064310(n) . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Mar 22 2007
%F A039599 T(2n,n)=A126596(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar
22 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*(-x)^k=A000007(n),A126983(n),A126984(n),A126982(n),
A126986(n),A126987(n),A127017(n),A127016(n),A126985(n),A127053(n)
for x=1,2,3,4,5,6,7,8,9,10 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 22 2007
%F A039599 Sum_{j, j>=0}T(n,j)*binomial(j,k)= A116395(n,k) . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Mar 30 2007
%F A039599 T(n,k)=Sum_{j, j>=0}A106566(n,j)*binomial(j,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 30 2007
%F A039599 T(n,k)=Sum_{j, j>=0}A127543(n,j)*A038207(j,k}. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Apr 03 2007
%F A039599 Sum_{k, 0<=k<=[n/2]}T(n-k,k)*A000108(k)=A101490(n+1). - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Apr 12 2007
%F A039599 T(n,k)=A053121(2*n,2*k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Apr 16 2007, Apr 17 2007, Apr 18 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*sin((2*k+1)*x)=sin(x)*(2*cos(x))^(2*n). - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Apr 16 2007, Apr 17 2007, Apr 18
2007
%F A039599 T(n,n-k)= Sum_{j, j>=0} (-1)^(n-j)*A094385(n,j)*binomial(j,k) . - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), May 05 2007
%F A039599 Sum_{j, j>=0}A110506(n,j)*binomial(j,k)=Sum_{j, j>=0}A110510(n,j)*A038207(j,
k)=T(n,k)*2^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May
25 2007
%F A039599 Sum_{j, j>=0}A110518(n,j)*A027465(j,k)=Sum_{j, j>=0}A110519(n,j)*A038207(j,
k)=T(n,k)*3^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May
25 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A001045(k)=A049027(n), for n>=1 . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Jun 09 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*a(k)=(m+2)^n if Sum_{k, k>=0}a(k)*x^k = (1+x)/
(x^2-m*x+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 09
2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A040000(k) = A001700(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jun 09 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A122553(k) = A051924(n+1) . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Jun 09 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A123932(k) = A051944(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jun 09 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*k^2 = A000531(n), for n>=1 . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Jun 10 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A000217(k)=A002457(n-1), for n>=1 . - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Jun 10 2007
%F A039599 Sum{j, j>=0}binomial(n,j)*T(j,k)= A124733(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Jun 16 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000984(n), A089022(n),
A035610(n), A130976(n), A130977(n), A130978(n), A130979(n), A130980(n),
A131521(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . -
Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A005043(k)=A127632(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 12 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A132262(k)=A089022(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 12 2007
%F A039599 T(n,k)+T(n,k+1)=A039598(n,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 12 2007
%F A039599 T(n,k) = A128899(n,k)+A128899(n,k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 12 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A015518(k) = A076025(n), for n>=1. Also Sum_{k,
0<=k<=n}T(n,k)*A015521(k) = A076026(n), for n>=1 . - Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Nov 22 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*x^(n-k) = A033999(n), A000007(n), A064062(n),
A110520(n), A132863(n), A132864(n), A132865(n), A132866(n), A132867(n),
A132869(n), A132897(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively
. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2007
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*(-1)^(k+1)*A000045(k)=A109262(n), A000045:= Fibonacci
numbers. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28
2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A000035(k)*A016116(k)=A143464(n). [From Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A016116(k)=A101850(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Oct 29 2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A010684(k)=A100320(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Oct 29 2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A000034(k)=A029651(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Oct 29 2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A010686(k)=A144706(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Oct 30 2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A006130(k-1)=A143646(n), with A006130(-1)=0 .
[From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 01 2008]
%F A039599 T(n,2*k)+T(n,2*k+1)=A118919(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 11 2008]
%F A039599 Sum_{k, 0<=k<=j}T(n,k)=A050157(n,j). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 12 2008]
%F A039599 Sum_{k, 0<=k<=2}T(n,k) = A026012(n); Sum_{k, 0<=k<=3}T(n,k)=A026029(n).
[From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 12 2008]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A000045(k+2)=A026671(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Feb 11 2009]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A000045(k+1)=A026726(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Feb 11 2009]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A057078(k)=A000012(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Feb 27 2009]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A108411(k)=A155084(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Mar 11 2009]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A057077(k)= 2^n = A000079(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Mar 11 2009]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*A057079(k) = 3^n = A000244(n). [From Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Mar 11 2009]
%F A039599 Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*A123344(k)= A000957(n+1). [From Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Nov 15 2009]
%e A039599 Triangle begins:
%e A039599 1;
%e A039599 1, 1;
%e A039599 2, 3, 1;
%e A039599 5, 9, 5, 1;
%e A039599 14, 28, 20, 7, 1;
%e A039599 42, 90, 75, 35, 9, 1;
%p A039599 T:=(n,k)->(2*k+1)*binomial(2*n,n-k)/(n+k+1): for n from 0 to 12 do seq(T(n,
k),k=0..n) od; # yields sequence in triangular form - Emeric Deutsch
(deutsch(AT)duke.poly.edu), May 06 2006
%Y A039599 Diagonals give : A000108 A000245 A000344 A000588 A001392 A000589 A000590,
A000012 A005408 A014107(n>1) Row sums : A000984
%Y A039599 Cf. A008313 A039598 A084938 A000007
%K A039599 nonn,tabl,easy,nice,new
%O A039599 0,4
%A A039599 N. J. A. Sloane (njas(AT)research.att.com).
%E A039599 More terms from Clark Kimberling (ck6(AT)evansville.edu)
%E A039599 Corrected by Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2009
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