A386771 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^3 / (1+3*x)^4 ).

1, 18, 474, 14732, 502401, 18180768, 685607224, 26650023732, 1060231986276, 42960995865518, 1766880793326474, 73566710202432732, 3094892737300954526, 131352228574805862768, 5617341984325110170724, 241825069451020881591732, 10471314920765093871735276

Offset: 0

Author

Seiichi Manyama, Aug 02 2025

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A386774.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^3/(1+3*x)^4)/x)

PARI

a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(4*(n+1), k)*binomial(4*n-k+2, n-k))/(n+1);

Formula

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

a(n) = (1/(n+1)) * [x^n] ( (1+3*x)^4 / (1-2*x)^3 )^(n+1).

D-finite with recurrence +168*(3*n+2)*(3*n+1)*(n+1)*a(n) +(-204763*n^3 +291562*n^2 -58913*n +7322)*a(n-1) +3*(2113057*n^3 -10391714*n^2 [phone]*n -5959810)*a(n-2) +15*(939475*n^3 [phone]*n^2 -61292611*n [phone])*a(n-3) -22987800*(2*n-5)*(4*n-9)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025