cp's OEIS Frontend

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)!.

%I A194558 #7 Mar 30 2012 18:37:28
%S A194558 1,1,2,3,11,15,88,115,893,1261,12226,16111,221227,282583,4411016,
%T A194558 6248747,113517609,148484297,3421012690,4385030203,110766993131,
%U A194558 153110987871,4175683922312,5442592336083,179150412103621,229026788618389,7917824064488690
%N 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)!.
%C A194558 This sequence is conjectured to consist entirely of integers.
%F A194558 a(n) = floor(n/2)!/n! * A194559(n).
%F A194558 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)!.
%e A194558 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! +...
%e A194558 The logarithm of the g.f. equals:
%e A194558 log(A(x)) = G_1(x) + G_2(x)^2/2 + G_3(x)^3/3 + G_4(x)^4/4 +...
%e A194558 where G_n(x) = x + x*G_n(x)^n is given by:
%e A194558 G_n(x) = Sum_{k>=0} C(n*k+1,k)/(n*k+1)*x^(n*k+1),
%e A194558 G_n(x)^n = Sum_{k>=1} C(n*k,k)/(n*k-k+1)*x^(n*k);
%e A194558 the first few expansions of G_n(x)^n begin:
%e A194558 G_1(x) = x + x^2 + x^3 + x^4 + x^5 +...
%e A194558 G_2(x)^2 = x^2 + 2*x^4 + 5*x^6 + 14*x^8 +...+ A000108(n)*x^(2*n) +...
%e A194558 G_3(x)^3 = x^3 + 3*x^6 + 12*x^9 + 55*x^12 +...+ A001764(n)*x^(3*n) +...
%e A194558 G_4(x)^4 = x^4 + 4*x^8 + 22*x^12 + 140*x^16 +...+ A002293(n)*x^(4*n) +...
%e A194558 G_5(x)^5 = x^5 + 5*x^10 + 35*x^15 + 285*x^20 +...+ A002294(n)*x^(5*n) +...
%o A194558 (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)}
%Y A194558  Cf. A194559.
%K A194558 nonn
%O A194558 0,3
%A A194558 _Paul D. Hanna_, Aug 28 2011