A390136 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x/(1+x)) ).

1, 1, 1, -2, -19, -24, 757, 6448, -29223, -1128320, -5570839, 193446144, 3805517509, -15555986432, -1721624631075, -18673954924544, 623030125550513, 22239755049664512, -28255798514535983, -18115342648598528000, -347257651967133027939, 10335451925606314541056

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Formula

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

E.g.f.: exp(B(x)), where B(x) is the e.g.f. of A334316.

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

a(n) = A243954(n+1)/(n+1).

Mathematica

Table[n!*Sum[(-1)^(n-k)*(n+1)^(k-1)*Binomial[n-1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Nov 05 2025 *)

Prog

(PARI) a(n, q=1, r=1, s=1, t=-1, u=0) = 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!);

Crossrefs

Cf. A243954, A334316, A361095.

Sequence in context: A186682 A031030 A083689 * A384997 A102617 A365235

Adjacent sequences: A390133 A390134 A390135 * A390137 A390138 A390139

Keyword

sign

Author

Seiichi Manyama, Oct 26 2025

Status

approved