A378483 Expansion of (Sum_{k>=0} binomial(3*k,k) * x^k)^3.

1, 9, 72, 549, 4077, 29772, 214884, 1537677, 10930923, 77298849, 544300992, 3819184236, 26718251868, 186440019192, 1298115301356, 9020928853341, 62582406445287, 433509545320731, 2998884192348888, 20720206275346269, 143005275737941437, 986000187782876976

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Jason Bland, Skip Garibaldi and Joel Rosenberg, Root of a cubic polynomial as a power series in the discriminant, arXiv:2605.25992 [math.GR], 2026.

Formula

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

G.f.: B(x)^3 where B(x) is the g.f. of A005809.

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

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

a(0) = 1; a(n) = (9/n) * Sum_{k=0..n-1} 2^k * binomial(k+3,3) * binomial(3*n+2,n-1-k). - Seiichi Manyama, May 03 2026

From Seiichi Manyama, May 05 2026: (Start)

a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(3*n+2,n-k).

a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(3*n-k,n-k).

a(0) = 1; a(n) = (9/n) * Sum_{k=0..n-1} 3^k * binomial(k+3,3) * binomial(3*n-2-k,n-1-k). (End)

Mathematica

nmax = 20; CoefficientList[Series[Sum[Binomial[3*k, k] * x^k, {k, 0, nmax}]^3, {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(3*k, k)*x^k)^3)

Crossrefs

Cf. A005809, A006256, A378504.

Cf. A002457, A378503.

Cf. A000302, A378484.

Sequence in context: A110396 A162755 A396209 * A045993 A084327 A057085

Adjacent sequences: A378480 A378481 A378482 * A378484 A378485 A378486

Keyword

nonn

Author

Seiichi Manyama, Nov 28 2024

Status

approved