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A183204 Central terms of triangle A181544. 3
1, 4, 48, 760, 13840, 273504, 5703096, 123519792, 2751843600, 62659854400, 1451780950048, 34116354472512, 811208174862904, 19481055861877120, 471822589361293680, 11511531876280913760, 282665135367572129040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The g.f. for row n of triangle A181544 is (1-x)^(3n+1)*Sum_{k>=0}C(n+k-1,k)^3*x^k.

This sequence is s_7 in Cooper's paper. - Jason Kimberley, Nov 06 2012

Diagonal of the rational function R(x,y,z,w)=1/(1-(w*x*y+w*x*z+w*y*z+x*y+x*z+y+z)). - Gheorghe Coserea, Jul 14 2016

LINKS

Jason Kimberley, Table of n, a(n) for n = 0..499

S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J., December 2012, Volume 29, Issue 1, pp 163-183.

Shaun Cooper, Jesús Guillera, Armin Straub, Wadim Zudilin, Crouching AGM, Hidden Modularity, arXiv:1604.01106 [math.NT], 5-April-2016.

Wadim Zudilin, A generating function of the squares of Legendre polynomials, preprint arXiv:1210.2493 [math.CA], 2012.

FORMULA

a(n) = [x^n] (1-x)^(3n+1) * Sum_{k>=0} C(n+k-1,k)^3*x^k.

a(n) = Sum_{j=0}^{n} C(n,j)^2 * C(2*j,n) * C(j+n,j). [Formula of Wadim Zudilin provided by Jason Kimberley, Nov 06 2012]

1/Pi = sqrt(7) Sum_{n>=0} (-1)^n a(n) (11895n+1286)/22^(3n+3). [Cooper, equation (41)] - Jason Kimberley, Nov 06 2012

G.f.: sqrt((1-13*x+(1-26*x-27*x^2)^(1/2))/(1-21*x+8*x^2+(1-8*x)*(1-26*x-27*x^2)^(1/2)))*hypergeom([1/12,5/12],[1],13824*x^7/(1-21*x+8*x^2+(1-8*x)*(1-26*x-27*x^2)^(1/2))^3)^2. - Mark van Hoeij, May 07 2013

a(n) ~ 3^(3*n+3/2) / (4 * (Pi*n)^(3/2)). - Vaclav Kotesovec, Apr 05 2015

G.f. A(x) satisfies 1/(1+4*x)^2 * A( x/(1+4*x)^3 ) = 1/(1+2*x)^2 * A( x^2/(1+2*x)^3 ) [see Cooper, Guillera, Straub, Zudilin]. - Joerg Arndt, Apr 08 2016

a(n)=(-1)^n*binomial(3n+1,n)* 4F3({-n,n+1,n+1,n+1};{1,1,2(n+1)}; 1). - M. Lawrence Glasser, May 15 2016

Conjecture: n^3*a(n) -(2*n-1) *(13*n^2-13*n+4)*a(n-1) -3*(n-1) *(3*n-4) *(3*n-2) *a(n-2)=0. - R. J. Mathar, May 15 2016

0 = (-x^2+26*x^3+27*x^4)*y'''  + (-3*x+117*x^2+162*x^3)*y'' + (-1+86*x+186*x^2)*y' + (4+24*x)*y, where y is g.f. -  Gheorghe Coserea, Jul 14 2016

EXAMPLE

Triangle A181544 begins:

(1);

1, (4), 1;

1, 20, (48), 20, 1;

1, 54, 405, (760), 405, 54, 1;

1, 112, 1828, 8464, (13840), 8464, 1828, 112, 1; ...

MATHEMATICA

Table[Sum[Binomial[n, j]^2 * Binomial[2*j, n] * Binomial[j+n, j], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 05 2015 *)

PROG

(PARI) {a(n)=polcoeff((1-x)^(3*n+1)*sum(j=0, 2*n, binomial(n+j, j)^3*x^j), n)}

(MAGMA) P<x>:=PolynomialRing(Integers()); C:=Binomial;

A183204:=func<n|Coefficient((1-x)^(3*n+1)*&+[C(n+j, j)^3*x^j:j in[0..2*n]], n)>; // or directly:

A183204:=func<k|&+[C(k, j)^2*C(2*j, k)*C(j+k, j):j in[0..k]]>;

[A183204(n):n in[0..16]]; // Jason Kimberley, Oct 29 2012

CROSSREFS

Cf. A181544, A183205.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A214819 A211198 A179235 * A047711 A089448 A167141

Adjacent sequences:  A183201 A183202 A183203 * A183205 A183206 A183207

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Dec 30 2010

STATUS

approved

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Last modified December 6 00:38 EST 2016. Contains 278771 sequences.