A380097 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - 3*x*exp(x)) ).

1, 3, 42, 1089, 42132, 2182335, 142084818, 11159447943, 1027313395944, 108517938075387, 12940759400071710, 1719811206219287643, 252076045285741340700, 40398758175398949144039, 7028240082095865121961514, 1319141702032289451776382975, 265703833060229155917857703888

Offset: 0

Author

Seiichi Manyama, Jan 12 2025

Keywords

• nonn

Crossrefs

Cf. A213644, A379688.

Cf. A380096.

Programs

PARI

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

Formula

E.g.f. A(x) satisfies A(x) = 1/( 1 - 3*x*A(x)*exp(x*A(x)) ).

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380096.

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

a(n) == 0 (mod 3) for n>0.