A178920 - OEIS

%I #21 Oct 30 2022 10:03:05

%S 1,4,39,616,13445,374976,12738523,510366592,23561390889,1231594508800,

%T 71902556218031,4637321353737216,327439395476545261,

%U 25123251004703358976,2081326422827575699875

%N Expansion of e.g.f. A(x), where A(x)=exp(x*A(x)+x^2*A(x)^2)

%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties </a>, arXiv:1103.2582 [math.CO], 2011-2013.

%F 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

%F 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

%t 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 *)

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Dec 29 2010