A369123 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2+x^2) ).

1, 2, 6, 20, 68, 224, 672, 1584, 880, -22880, -215072, -1414400, -8012032, -41344000, -198120448, -884348160, -3640426752, -13403384320, -40424947200, -65476561920, 329862128640, 4603911045120, 35276325027840, 221747978649600, 1244854463643648

Offset: 0

Author

Seiichi Manyama, Jan 13 2024

Keywords

• sign

Links

• Index entries for reversions of series

Crossrefs

Cf. A097188, A369124, A369125.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2+x^2))/x)

PARI

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

Formula

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