A378484 Expansion of (Sum_{k>=0} binomial(4*k,k) * x^k)^4.

1, 16, 208, 2480, 28176, 310336, 3344688, 35472672, 371570320, 3853862080, 39650662720, 405221752112, 4117879215472, 41643345090240, 419362920305952, 4207604570770752, 42079232716865424, 419609034657373120, 4173470598366784960, 41413032430984848832, 410071444666659404352

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 0} binomial(4*i,i) * binomial(4*j,j) * binomial(4*k,k) * binomial(4*l,l).

G.f.: B(x)^4 where B(x) is the g.f. of A005810.

27*a(n) - 256*a(n-1) = 18*A078995(n) + 8*A005810(n) for n > 0.

a(n) ~ n * 2^(8*n + 2) / 3^(3*n + 2) * (1 + 2^(7/2)/(3^(3/2)*sqrt(Pi*n))). - Vaclav Kotesovec, Jul 19 2025

a(0) = 1; a(n) = (16/n) * Sum_{k=0..n-1} 3^k * binomial(k+4,4) * binomial(4*n+3,n-1-k). - Seiichi Manyama, May 03 2026

From Seiichi Manyama, May 05 2026: (Start)

a(n) = Sum_{k=0..n} 3^k * binomial(k+2,2) * binomial(4*n+3,n-k).

a(n) = Sum_{k=0..n} 4^k * binomial(k+2,2) * binomial(4*n-k,n-k).

a(0) = 1; a(n) = (16/n) * Sum_{k=0..n-1} 4^k * binomial(k+4,4) * binomial(4*n-2-k,n-1-k). (End)

Mathematica

nmax = 20; CoefficientList[Series[Sum[Binomial[4*k, k] * x^k, {k, 0, nmax}]^4, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 19 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(4*k, k)*x^k)^4)

Crossrefs

Cf. A005810, A078995, A378503, A395659.

Cf. A000302, A378483.

Sequence in context: A055758 A046088 A186853 * A016159 A297370 A285167

Adjacent sequences: A378481 A378482 A378483 * A378485 A378486 A378487

Keyword

nonn

Author

Seiichi Manyama, Nov 28 2024

Status

approved