A370194 Coefficient of x^n in the expansion of ( (1+x) * (1+x^2)^2 )^n.

1, 1, 5, 19, 77, 326, 1391, 6028, 26349, 116011, 513730, 2285570, 10208111, 45742724, 205550840, 925918544, 4179740909, 18903381337, 85635147983, 388517336189, 1765019420602, 8028115465732, 36555667019338, 166621503161184, 760161934681647, 3470945792364701

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(n,n-2*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) * (1+x^2)^2) ). See A369440.

Mathematica

a[n_]:=SeriesCoefficient[((1+x)*(1+x^2)^2)^n, {x, 0, n}]; Array[a, 26, 0] (* Stefano Spezia, Apr 30 2024 *)

Prog

(PARI) a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial(u*n, n-s*k));

Crossrefs

Cf. A369440, A370159.

Sequence in context: A228678 A149771 A149772 * A149773 A363548 A149774

Adjacent sequences: A370191 A370192 A370193 * A370195 A370196 A370197

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 11 2024

Status

approved