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E.g.f.: Series_Reversion( x*Cw(x) ) where Cw(x) = Sum_{n>=0} (-1)^n*(2*n+1)^(2*n-1)*x^(2*n)/(2*n)!.
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%I #10 Sep 08 2012 00:14:55

%S 1,9,185,6769,384849,31247161,3421948361,485057489505,86270172949025,

%T 18789108183911401,4913945007420622425,1518613513007413125073,

%U 547156929866111948071025,227227144424871839232479769,107701858026047543489146771049

%N E.g.f.: Series_Reversion( x*Cw(x) ) where Cw(x) = Sum_{n>=0} (-1)^n*(2*n+1)^(2*n-1)*x^(2*n)/(2*n)!.

%F E.g.f. A(x) satisfies:

%F (1) Sum_{n>=0} (-1)^n*(2*n+1)^(2*n)*A(x)^(2*n+1)/(2*n+1)! = x.

%F (2) A( atan(Sw(x)/Cw(x)) ) = x where Sw(x) = Sum_{n>=0} (-1)^n*(2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.

%e E.g.f.: A(x) = x + 9*x^3/3! + 185*x^5/5! + 6769*x^7/7! + 384849*x^9/9! +...

%e such that A(x*Cw(x)) = x where

%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 Related expansion:

%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 where Cw(x) + I*Sw(x) = LambertW(-I*x)/(-I*x).

%o (PARI) {a(n)=local(Cw=sum(m=0,n,(-1)^m*(2*m+1)^(2*m-1)*x^(2*m)/(2*m)!) +x*O(x^n));n!*polcoeff(serreverse(x*Cw),n)}

%o for(n=1,20,print1(a(2*n-1),", ")) \\ print only odd-indexed terms

%Y Cf. A215890, A138734, A215880, A215881, A215882, A216143.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 06 2012