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

1, 1, 4, 18, 132, 900, 10560, 96600, 1500240, 19066320, 369714240, 5359163040, 147177898560, 2443958637120, 76298578836480, 1621294897622400, 58906376034105600, 1309870975014201600, 60357698670132864000, 1469955465552513139200, 74262907856067567436800

Offset: 0

Author

Paul D. Hanna, Aug 28 2011

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 132*x^4/4! + 900*x^5/5! +...
The logarithm of the g.f. equals:
log(A(x)) = G_1(x) + G_2(x)^2/2 + G_3(x)^3/3 + G_4(x)^4/4 +...
where G_n(x) = x + x*G_n(x)^n is given by:
G_n(x) = Sum_{k>=0} C(n*k+1,k)/(n*k+1)*x^(n*k+1),
G_n(x)^n = Sum_{k>=1} C(n*k,k)/(n*k-k+1)*x^(n*k);
the first few expansions of G_n(x)^n begin:
G_1(x) = x + x^2 + x^3 + x^4 + x^5 +...
G_2(x)^2 = x^2 + 2*x^4 + 5*x^6 + 14*x^8 +...+ A000108(n)*x^(2*n) +...
G_3(x)^3 = x^3 + 3*x^6 + 12*x^9 + 55*x^12 +...+ A001764(n)*x^(3*n) +...
G_4(x)^4 = x^4 + 4*x^8 + 22*x^12 + 140*x^16 +...+ A002293(n)*x^(4*n) +...
G_5(x)^5 = x^5 + 5*x^10 + 35*x^15 + 285*x^20 +...+ A002294(n)*x^(5*n) +...
		

Crossrefs

Cf. A194558, A194560.

Programs

PARI

{a(n)=n!*polcoeff(exp(sum(m=1,n+1,serreverse(x/(1+x^m+x*O(x^n)))^m/m)),n)}

Formula

a(n) = n!/floor(n/2)! * A194558(n).

E.g.f.: A(x) = exp( Sum_{n>=1} Series_Reversion( x/(1+x^n) )^n/n ).