A378484 - OEIS

%I #34 May 05 2026 09:23:57

%S 1,16,208,2480,28176,310336,3344688,35472672,371570320,3853862080,

%T 39650662720,405221752112,4117879215472,41643345090240,

%U 419362920305952,4207604570770752,42079232716865424,419609034657373120,4173470598366784960,41413032430984848832,410071444666659404352

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

%H Seiichi Manyama, <a href="/A378484/b378484.txt">Table of n, a(n) for n = 0..1000</a>

%F 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).

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

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

%F 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

%F 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

%F From _Seiichi Manyama_, May 05 2026: (Start)

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

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

%F 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)

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

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

%Y Cf. A005810, A078995, A378503, A395659.

%Y Cf. A000302, A378483.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 28 2024