A385320 a(n) = Sum_{k=0..n} 2^k * binomial(3*n,k) * binomial(3*n-k-1,n-k).

1, 8, 118, 1970, 34714, 630548, 11678284, 219240008, 4157096266, 79429466456, 1526869550638, 29495424821354, 572100064904872, 11134578632483600, 217341014671302976, 4253067310380772400, 83409477100625759050, 1638952453699219007072, 32259670449587082804466

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = [x^n] ( (1+2*x)^3/(1-x)^2 )^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^2 / (1+2*x)^3 ). See A386722.

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

a(n) ~ 3^(4*n + 3/2) / (7*sqrt(Pi*n)*2^(2*n)). - Vaclav Kotesovec, Jul 31 2025

a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n+k-1,k). - Seiichi Manyama, Aug 01 2025

Mathematica

Table[Sum[3^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Jul 31 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*binomial(3*n, k)*binomial(3*n-k-1, n-k));

Crossrefs

Cf. A385319, A386719.

Cf. A386722.

Sequence in context: A342182 A194494 A230911 * A267752 A386365 A318009

Adjacent sequences: A385317 A385318 A385319 * A385321 A385322 A385323

Keyword

nonn

Author

Seiichi Manyama, Jul 31 2025

Status

approved