A216409 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)!.

1, 9, 185, 6769, 384849, 31247161, 3421948361, 485057489505, 86270172949025, 18789108183911401, 4913945007420622425, 1518613513007413125073, 547156929866111948071025, 227227144424871839232479769, 107701858026047543489146771049

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

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

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

(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)!.

Example

E.g.f.: A(x) = x + 9*x^3/3! + 185*x^5/5! + 6769*x^7/7! + 384849*x^9/9! +...
such that A(x*Cw(x)) = x where
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)! +...
Related expansion:
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)! +...
where Cw(x) + I*Sw(x) = LambertW(-I*x)/(-I*x).

Prog

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

Crossrefs

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

Sequence in context: A319798 A300598 A189803 * A171194 A196297 A274781

Adjacent sequences: A216406 A216407 A216408 * A216410 A216411 A216412

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2012

Status

approved