A386866 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+2,k) * binomial(3*n-k-1,n-k).

1, 9, 132, 2197, 38649, 701292, 12979360, 243541725, 4616122851, 88173726337, 1694554311888, 32728267058604, 634701136059532, 12351249029265816, 241061116082196072, 4716751239386395885, 92494719333403946583, 1817328001770278062299, 35768122814759119268788

Offset

0,2

Links

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

Formula

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

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

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

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

Mathematica

Table[Sum[2^(n-k) Binomial[3n+2, k]Binomial[3n-k-1, n-k], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Sep 02 2025 *)

Prog

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

Crossrefs

Cf. A384950, A386863.

Cf. A386836.

Sequence in context: A282820 A296318 A167253 * A366017 A097999 A089547

Adjacent sequences: A386863 A386864 A386865 * A386867 A386868 A386869

Keyword

nonn

Author

Seiichi Manyama, Aug 06 2025

Status

approved