cp's OEIS Frontend

A194559 E.g.f.: exp( Sum_{n>=1} G_n(x)^n/n ) where G_n(x) = x + x*G_n(x)^n.

%I A194559 #8 Mar 30 2012 18:37:28
%S A194559 1,1,4,18,132,900,10560,96600,1500240,19066320,369714240,5359163040,
%T A194559 147177898560,2443958637120,76298578836480,1621294897622400,
%U A194559 58906376034105600,1309870975014201600,60357698670132864000,1469955465552513139200,74262907856067567436800
%N A194559 E.g.f.: exp( Sum_{n>=1} G_n(x)^n/n ) where G_n(x) = x + x*G_n(x)^n.
%F A194559 a(n) = n!/floor(n/2)! * A194558(n).
%F A194559 E.g.f.: A(x) = exp( Sum_{n>=1} Series_Reversion( x/(1+x^n) )^n/n ).
%e A194559 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 132*x^4/4! + 900*x^5/5! +...
%e A194559 The logarithm of the g.f. equals:
%e A194559 log(A(x)) = G_1(x) + G_2(x)^2/2 + G_3(x)^3/3 + G_4(x)^4/4 +...
%e A194559 where G_n(x) = x + x*G_n(x)^n is given by:
%e A194559 G_n(x) = Sum_{k>=0} C(n*k+1,k)/(n*k+1)*x^(n*k+1),
%e A194559 G_n(x)^n = Sum_{k>=1} C(n*k,k)/(n*k-k+1)*x^(n*k);
%e A194559 the first few expansions of G_n(x)^n begin:
%e A194559 G_1(x) = x + x^2 + x^3 + x^4 + x^5 +...
%e A194559 G_2(x)^2 = x^2 + 2*x^4 + 5*x^6 + 14*x^8 +...+ A000108(n)*x^(2*n) +...
%e A194559 G_3(x)^3 = x^3 + 3*x^6 + 12*x^9 + 55*x^12 +...+ A001764(n)*x^(3*n) +...
%e A194559 G_4(x)^4 = x^4 + 4*x^8 + 22*x^12 + 140*x^16 +...+ A002293(n)*x^(4*n) +...
%e A194559 G_5(x)^5 = x^5 + 5*x^10 + 35*x^15 + 285*x^20 +...+ A002294(n)*x^(5*n) +...
%o A194559 (PARI) {a(n)=n!*polcoeff(exp(sum(m=1,n+1,serreverse(x/(1+x^m+x*O(x^n)))^m/m)),n)}
%Y A194559 Cf. A194558, A194560.
%K A194559 nonn
%O A194559 0,3
%A A194559 _Paul D. Hanna_, Aug 28 2011