A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

1, 5, 20, 95, 400, 1561, 5915, 21610, 76585, 265075, 898622, 2992235, 9810290, 31727815, 101379175, 320464280, 1003259080, 3113576320, 9586763720, 29305985800, 88997753446, 268642069750, 806394498200, 2408144329250, 7157177344225, 21177323087891

Offset: 0

Author

Seiichi Manyama, Oct 18 2024

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..2357

Crossrefs

Cf. A051286, A182884, A377145, A377148, A377153, A377158, A377159.

Cf. A001873, A089627.

Programs

Maple

f:= proc(n) local k; add(binomial(k+4,4)*binomial(k,n-k)^2,k=0..n) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024

PARI

a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(k, n-k)^2);

PARI

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: (Sum_{k=0..2} A089627(4,k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(9/2).