A380664 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)^2) ).

1, 2, 17, 268, 6277, 196416, 7716109, 365398496, 20271580137, 1290027358720, 92653747607401, 7414981595716608, 654373744057368493, 63136350047908917248, 6612064512998173129125, 747016321343021395603456, 90564758322246657646854481, 11727981253987656671672008704

Offset

0,2

Links

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

Index entries for reversions of series

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x))^2)/(1 - x*A(x)).

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

a(n) = binomial(2*n, n)*n!*hypergeom([-n, 1+2*n], [(1+n)/2, 1+n/2], -(1+n)/4)/(1 + n). - Stefano Spezia, Nov 05 2025

Mathematica

Table[n!*Sum[(n+1)^(k-1)*Binomial[2*n+k, n-k]/k!, {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Nov 05 2025 *)
a[n_]:=Binomial[2n, n]*n!*HypergeometricPFQ[{-n, 1+2*n}, {(1+n)/2, 1+n/2}, -(1+n)/4]/(1 + n); Array[a, 18, 0] (* Stefano Spezia, Nov 05 2025 *)

Prog

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

Crossrefs

Cf. A377831, A380663.

Cf. A361598.

Sequence in context: A301584 A387967 A195443 * A176585 A086534 A198287

Adjacent sequences: A380661 A380662 A380663 * A380665 A380666 A380667

Keyword

nonn

Author

Seiichi Manyama, Jan 30 2025

Status

approved