%I #8 Sep 08 2012 00:12:37
%S 1,4,79,3872,357021,53366688,11788384035,3613002977280,
%T 1467889838452377,763713003999744000,495264178234423963575,
%U 391720087063508887535616,371190938737957616525807925,415169544652854511226963558400,541213248030886833323809041196875
%N E.g.f.: Series_Reversion( 2*Sw(x/2) ) where Sw(x) = Sum_{n>=0} (-1)^n*(2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
%F E.g.f. A(x) satisfies: Sum_{n>=0} (-1)^n*(n+1)^(2*n) * A(x)^(2*n+1)/(2*n+1)! = x.
%e E.g.f.: A(x) = x + 4*x^3/3! + 79*x^5/5! + 3872*x^7/7! + 357021*x^9/9! +...
%e such that A(2*Sw(x/2)) = x, where
%e 2*Sw(x/2) = x - 4*x^3 + 81*x^5 - 4096*x^7 + 390625*x^9 - 60466176*x^11 +...+ (-1)^n*(n+1)^(2*n)*x^(2*n+1)/(2*n+1)! +...
%e Related expansions:
%e Sw(x) = x - 16*x^3/3! + 1296*x^5/5! - 262144*x^7/7! + 100000000*x^9/9! -+...+ (-1)^n*(2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...
%e Cw(x) = 1 - 3*x^2/2! + 125*x^4/4! - 16807*x^6/6! + 4782969*x^8/8! -+...+ (-1)^n*(2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...
%e where Cw(x) + I*Sw(x) = LambertW(-I*x)/(-I*x).
%o (PARI) {a(n)=local(A=x);A=serreverse(sum(m=0,n\2,(-1)^m*(m+1)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n));n!*polcoeff(A,n)}
%o for(n=1,21,print1(a(2*n-1),", ")) \\ print only odd-indexed terms
%Y Cf. A216409.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 08 2012