A369301 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^3)^3 ).

1, 3, 15, 94, 657, 4902, 38236, 308025, 2542965, 21401780, 182934144, 1583745114, 13858675065, 122379042879, 1089156646584, 9759520978270, 87975115569873, 797233088237190, 7258632128721117, 66367727370376632, 609132332475784548, 5610015849998778144

Offset: 0

Author

Seiichi Manyama, Jan 18 2024

Keywords

• nonn

Links

• Peter Bala, Fractional iteration of a series inversion operator
• Index entries for reversions of series

Crossrefs

Cf. A369299, A369300.

Cf. A369270.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*(1-x^3)^3)/x)

PARI

a(n, s=3, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+k+2,k) * binomial(4*n-3*k+2,n-3*k).

a(n) = (1/(n+1)) * [x^n] 1/( (1-x)^3 * (1-x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024