A178920 E.g.f. A(x) satisfies A(x) = x * (1 + A(x)) * exp( A(x) * (1 + A(x)) ).

1, 4, 39, 616, 13445, 374976, 12738523, 510366592, 23561390889, 1231594508800, 71902556218031, 4637321353737216, 327439395476545261, 25123251004703358976, 2081326422827575699875, 185163079403113846767616, 17606094731689771913952977, 1781830510631928735743213568

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Index entries for reversions of series

Formula

a(n) = (n-1)!*Sum_{i=0..n-1} binomial(n+i,n-i-1)*n^i/i!. - corrected by Vaclav Kotesovec, Jan 26 2014

a(n) ~ c * n^n * ( exp(m-1) * (1+m)^(1+m) / (m^(3*m) * 2^(2*m) * (1-m)^(1-m)) )^n, where m = 1/12*(-1 + (215 - 12*sqrt(321))^(1/3) + (215 + 12*sqrt(321))^(1/3)) = 0.5566930950324... is the root of the equation m^2*(1+4*m)=1, and c = 2.194433179699246977948075450550764549... - Vaclav Kotesovec, Jan 26 2014

E.g.f.: Series_Reversion( x * exp(-x * (1 + x)) / (1 + x) ). - Seiichi Manyama, Feb 03 2025

Mathematica

a[n_] := (n+1)!*HypergeometricPFQ[ {-n, n+2}, {1, 3/2}, -(n+1)/4]; Table[a[n], {n, 0, 14}]  (* Jean-François Alcover, Mar 01 2013 *)

Prog

(PARI) a(n) = (n-1)!*sum(k=0, n-1, n^k*binomial(n+k, n-1-k)/k!); \\ Seiichi Manyama, Feb 03 2025

Crossrefs

Cf. A088695.

Sequence in context: A365010 A024055 A361544 * A066399 A065760 A132612

Adjacent sequences: A178917 A178918 A178919 * A178921 A178922 A178923

Keyword

nonn

Author

Vladimir Kruchinin, Dec 29 2010

Extensions

Name and offset corrected by Seiichi Manyama, Feb 03 2025

Status

approved