A372476 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x-x^3) )^n.

1, 2, 10, 59, 366, 2332, 15127, 99388, 659262, 4405379, 29611120, 199986085, 1356018339, 9225340880, 62941829996, 430495159084, 2950754125870, 20263845589461, 139393311839827, 960318328961614, 6624842357972916, 45757925847607270, 316401673996278705, 2190032296476131430

Offset

0,2

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,k) * binomial(3*n-2*k-1,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) * (1-x-x^3) ). See A368931.

Mathematica

a[n_]:=SeriesCoefficient[1 / ( (1-x) * (1-x-x^3) )^n, {x, 0, n}]; Array[a, 23, 0] (* Stefano Spezia, Jun 02 2026*)

Prog

(PARI) a(n, s=3, t=1, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t+u+1)*n-(s-1)*k-1, n-s*k));

Crossrefs

Cf. A368931, A372233.

Sequence in context: A340987 A186758 A372415 * A262910 A370281 A351502

Adjacent sequences: A372473 A372474 A372475 * A372477 A372478 A372479

Keyword

nonn

Author

Seiichi Manyama, May 02 2024

Extensions

a(23) from Stefano Spezia, Jun 02 2026

Status

approved