A387967 E.g.f. A(x) satisfies A(x) = exp( x*A(x) * (1+x*A(x))^3 ) * (1+x*A(x)).

1, 2, 17, 268, 6193, 190776, 7370089, 342965456, 18688659777, 1167774941440, 82336991069761, 6467576529650688, 560146953164631793, 53034384052597326848, 5449833727606226158425, 604111482231039865747456, 71856281544136203151468801, 9129117145477837345927593984

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

E.g.f.: (1/x) * Series_Reversion( x * exp(-x * (1+x)^3) / (1+x) ).

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

Mathematica

Table[n!*Sum[(n+1)^(k-1)*Binomial[n+3*k+1, n-k]/k!, {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Oct 27 2025 *)

Prog

(PARI) a(n, q=1, r=1, s=1, t=3, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial(r*u*n+((s-r)*u+t)*k+q*u, n-k)/k!);
(Magma) [Factorial(n) * &+[(n+1)^(k-1)* Binomial(n+3*k+1, n-k) / Factorial(k) : k in [0..n]] : n in [0..25] ]; // Vincenzo Librandi, Oct 27 2025

Crossrefs

Cf. A178920, A377964, A380762.

Sequence in context: A099714 A379847 A301584 * A195443 A380664 A176585

Adjacent sequences: A387964 A387965 A387966 * A387968 A387969 A387970

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2025

Status

approved