A371816 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(3*n-3*k-1,n-3*k).

1, 2, 10, 55, 322, 1947, 12013, 75154, 474946, 3024742, 19381045, 124797862, 806875421, 5234713031, 34060165282, 222174355575, 1452425614146, 9513309908589, 62418283102246, 410161124310550, 2698932409666237, 17781425199962255, 117281204608676426

Offset: 0

Author

Seiichi Manyama, Apr 06 2024

Keywords

• nonn

Crossrefs

Cf. A120305, A165817, A371817.

Cf. A371770.

Programs

Mathematica

Table[Sum[(-1)^k Binomial[3n-3k-1,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Aug 07 2025 *)

PARI

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

Formula

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

a(n) = binomial(3*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/3-n, 2/3-n, 1-n], -1). - Stefano Spezia, Apr 07 2024