A380663 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)) ).

1, 2, 15, 208, 4285, 117936, 4075099, 169736960, 8282604537, 463604723200, 29287449579751, 2061571190059008, 160023548976361525, 13580237335641417728, 1250935473495646861875, 124307671411309327876096, 13255531892787507819759601, 1509841440567809574906101760

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Index entries for reversions of series

Formula

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

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

a(n) ~ 2^(n - 1/2) * (1 + sqrt(2))^(n + 1/2) * n^(n-1) / exp((2 - sqrt(2))*n + 1 - sqrt(2)). - Vaclav Kotesovec, Feb 01 2026

Mathematica

Table[n! * Sum[(n+1)^(k-1) * Binomial[2*n, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 01 2026 *)

Prog

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

Crossrefs

Cf. A377831, A380664.

Cf. A052873, A380665.

Sequence in context: A099718 A143881 A380762 * A377890 A298692 A361617

Adjacent sequences: A380660 A380661 A380662 * A380664 A380665 A380666

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2025

Status

approved