A378504 Expansion of (Sum_{k>=0} binomial(3*k,k) * x^k)^4.

1, 12, 114, 984, 8055, 63744, 492702, 3742704, 28053423, 208057260, 1529802648, 11168142048, 81041199876, 585045970992, 4204705925670, 30101448952032, 214756404746031, 1527491122906212, 10834911076417458, 76666402505673720, 541277205506059743

Offset: 0

Author

Seiichi Manyama, Nov 28 2024

Keywords

• nonn

Links

• Marko Riedel et al, Identity between two generating functions

Crossrefs

Cf. A005809, A006256, A378483.

Cf. A002697, A378484.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Sum[Binomial[3*k, k] * x^k, {k, 0, nmax}]^4, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 23 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(3*k, k)*x^k)^4)

Formula

a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 0} binomial(3*i,i) * binomial(3*j,j) * binomial(3*k,k) * binomial(3*l,l).

G.f.: B(x)^4 where B(x) is the g.f. of A005809.

4*a(n) - 27*a(n-1) = 3*A006256(n) + A005809(n) for n > 0.

Sum_{n >= 0} a(n) * z^n / (1+z)^(3*n) = (1+z)^4 / (1-2*z)^4. - Marko Riedel, Jul 22 2025

From Vaclav Kotesovec, Jul 23 2025: (Start)

Recurrence: 8*(n-1)*n*(2*n - 1)*a(n) = 6*(n-1)*(36*n^2 - 9*n - 5)*a(n-1) - 81*n*(3*n - 2)*(3*n - 1)*a(n-2).

a(n) ~ n * 3^(3*n+2) / 2^(2*n+4). (End)