A216409 Expansion of e.g.f. (x/cos(x)) * exp(x*tan(x)) (odd powers only).

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

Offset: 1

Author

Paul D. Hanna, Sep 06 2012

Keywords

• nonn

Comments

Name changed Mar 24 2025; prior name was: e.g.f. Series_Reversion( x*Cw(x) ), which is equivalent to formula (2).

Examples

			E.g.f.: A(x) = x + 9*x^3/3! + 185*x^5/5! + 6769*x^7/7! + 384849*x^9/9! + ...
Related expansions.
Define Cw(x) and Sw(x) by Cw(x) + i*Sw(x) = LambertW(-i*x)/(-i*x), then
Cw(x) + i*Sw(x) = (cos(x*Cw(x)) + i*sin(x*Cw(x))) * exp(-x*Sw(x)) and
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)! + ...
where A(x*Cw(x)) = x, and
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 A( arctan(Sw(x)/Cw(x)) ) = x.
Also, Sw(x)/Cw(x) = tan( x*Cw(x) ) and begins
Sw(x)/Cw(x) = x - 7*x^3/3! + 461*x^5/5! - 84827*x^7/7! + 30648697*x^9/9! - 18319873199*x^11/11! +- ... + (-1)^(n-1) * A215890(n)*x^(2*n-1) + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..201

Crossrefs

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

Programs

Mathematica

nmax = 20; Quiet[Rest[CoefficientList[Series[x*E^(Sqrt[x]*Tan[Sqrt[x]])*Sec[Sqrt[x]], {x, 0, nmax}], x] * (2*Range[0, nmax]-1)!]] (* Vaclav Kotesovec, Mar 24 2025 *)

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

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! satisfies:

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

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

(3) A( arctan(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)!.

(4) A(x) = (x/cos(x)) * exp(x*tan(x)). - Paul D. Hanna, Mar 24 2025

a(n) ~ 2^(4*n - 5/4) * n^(2*n - 3/4) / (Pi^(2*n-1) * exp(2*n - 2*sqrt(2*n) + 1/2)) * (1 - (14 + Pi^2)/(12*sqrt(2*n))). - Vaclav Kotesovec, Mar 24 2025

Extensions

Name changed and entry revised by Paul D. Hanna, Mar 24 2025