A386900 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+1,k).

1, 14, 235, 4178, 76495, 1426184, 26922076, 512838410, 9837067951, 189729498350, 3675700225435, 71474375851640, 1394164222173700, 27266825345422352, 534510606516137920, 10499123975453808698, 206595710100771337327, 4071693103719194746250

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

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

a(n) = [x^n] 1/((1-3*x)^(2*n+1) * (1-5*x)).

a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).

a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(3*n-k,n-k).

a(n) ~ 3^(4*n + 5/2) / (sqrt(Pi*n) * 2^(2*n+3)). - Vaclav Kotesovec, Aug 07 2025

Mathematica

Table[Sum[3^k*2^(n-k)*Binomial[3*n+1, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *)

Prog

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

Crossrefs

Cf. A386830, A386899.

Sequence in context: A280559 A305862 A222377 * A220502 A256462 A166774

Adjacent sequences: A386897 A386898 A386899 * A386901 A386902 A386903

Keyword

nonn

Author

Seiichi Manyama, Aug 07 2025

Status

approved