A386770 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+3*x)^3 ).

1, 13, 244, 5397, 130961, 3372268, 90497184, 2503434117, 70883043571, 2044268649573, 59842331451024, 1773506049794412, 53107658756034156, 1604418047921589928, 48841208603255888264, 1496711470907670605157, 46134317696761847385591, 1429405788411234205692583

Offset: 0

Author

Seiichi Manyama, Aug 02 2025

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A385474, A386764.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+3*x)^3)/x)

PARI

a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

Formula

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

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

D-finite with recurrence 32*(n+1)*(2*n+1)*a(n) +48*(-81*n^2+27*n-7)*a(n-1) +162*(414*n^2-891*n+605)*a(n-2) -32805*(3*n-4)*(3*n-5)*a(n-3)=0. - R. J. Mathar, Aug 03 2025