A370285 Coefficient of x^n in the expansion of ( (1+x)^2 + x^3 )^n.

1, 2, 6, 23, 94, 392, 1659, 7107, 30734, 133880, 586576, 2582142, 11411371, 50597900, 224986467, 1002867878, 4479814606, 20049099908, 89878609344, 403521966942, 1814102538624, 8165526187128, 36794746597494, 165968135843522, 749314496125451, 3385881647958442

Offset

0,2

Links

Table of n, a(n) for n=0..25.

Formula

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

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 + x^3) ). See A369212.

Maple

a := n -> binomial(2*n, n) * hypergeom([(1-n)/3, (2-n)/3, -n/3], [1/2-n, n+1], 27/4):
seq(simplify(a(n)), n = 0..25);  # Peter Luschny, Jan 04 2025

Prog

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

Crossrefs

Cf. A006139, A026375, A369212.

Similar to A082759.

Sequence in context: A150292 A191721 A150293 * A371827 A150294 A150295

Adjacent sequences: A370282 A370283 A370284 * A370286 A370287 A370288

Keyword

nonn

Author

Seiichi Manyama, Feb 14 2024

Status

approved