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

%I A216409 #35 Mar 26 2025 16:36:08
%S A216409 1,9,185,6769,384849,31247161,3421948361,485057489505,86270172949025,
%T A216409 18789108183911401,4913945007420622425,1518613513007413125073,
%U A216409 547156929866111948071025,227227144424871839232479769,107701858026047543489146771049
%N A216409 Expansion of e.g.f. (x/cos(x)) * exp(x*tan(x)) (odd powers only).
%C A216409 Name changed Mar 24 2025; prior name was: e.g.f. Series_Reversion( x*Cw(x) ), which is equivalent to formula (2).
%H A216409 Paul D. Hanna, <a href="/A216409/b216409.txt">Table of n, a(n) for n = 1..201</a>
%F A216409 E.g.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! satisfies:
%F A216409 (1) Sum_{n>=0} (-1)^n * (2*n+1)^(2*n) * A(x)^(2*n+1)/(2*n+1)! = x.
%F A216409 (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)!.
%F A216409 (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)!.
%F A216409 (4) A(x) = (x/cos(x)) * exp(x*tan(x)). - _Paul D. Hanna_, Mar 24 2025
%F A216409 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
%e A216409 E.g.f.: A(x) = x + 9*x^3/3! + 185*x^5/5! + 6769*x^7/7! + 384849*x^9/9! + ...
%e A216409 Related expansions.
%e A216409 Define Cw(x) and Sw(x) by Cw(x) + i*Sw(x) = LambertW(-i*x)/(-i*x), then
%e A216409 Cw(x) + i*Sw(x) = (cos(x*Cw(x)) + i*sin(x*Cw(x))) * exp(-x*Sw(x)) and
%e A216409 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 A216409 where A(x*Cw(x)) = x, and
%e A216409 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 A216409 where A( arctan(Sw(x)/Cw(x)) ) = x.
%e A216409 Also, Sw(x)/Cw(x) = tan( x*Cw(x) ) and begins
%e A216409 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) + ...
%t A216409 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 *)
%o A216409 (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 A216409 for(n=1,20,print1(a(2*n-1),", ")) \\ print only odd-indexed terms
%Y A216409 Cf. A215890, A138734, A215880, A215881, A215882, A216143.
%K A216409 nonn
%O A216409 1,2
%A A216409 _Paul D. Hanna_, Sep 06 2012
%E A216409 Name changed and entry revised by _Paul D. Hanna_, Mar 24 2025