A194558 G.f.: A(x) = exp( Sum_{n>=1} G_n(x)^n/n ) where G_n(x) = x + x*G_n(x)^n and A(x) = Sum_{n>=1} a(n)*x^n/floor(n/2)!.
1, 1, 2, 3, 11, 15, 88, 115, 893, 1261, 12226, 16111, 221227, 282583, 4411016, 6248747, 113517609, 148484297, 3421012690, 4385030203, 110766993131, 153110987871, 4175683922312, 5442592336083, 179150412103621, 229026788618389, 7917824064488690
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 11*x^4/2! + 15*x^5/2! + 88*x^6/3! + 115*x^7/3! + 893*x^8/4! + 1261*x^9/4! + 12226*x^10/5! + 16111*x^11/5! +...
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. A194559.
Programs
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PARI
{a(n)=floor(n/2)!*polcoeff(exp(sum(m=1,n+1,serreverse(x/(1+x^m+x*O(x^n)))^m/m)),n)}
Formula
a(n) = floor(n/2)!/n! * A194559(n).
G.f.: A(x) = exp( Sum_{n>=1} Series_Reversion( x/(1+x^n) )^n/n ) where A(x) = Sum_{n>=1} a(n)*x^n/floor(n/2)!.
Comments