A184972 - OEIS

%I #21 Mar 01 2024 02:04:47

%S 1,1,3,13,81,605,5595,59225,725985,9928505,151720275,2541096325,

%T 46541735025,922017392725,19691502952875,450278539452625,

%U 10987846186994625,284800630720672625,7817729823142243875,226487095510937568125,6907505385375525620625

%N Expansion of e.g.f. 1/( cos(arctanh(x)) - sin(arctanh(x)) ).

%C Compare e.g.f. to 1/(cosh(arctanh(x)) - sinh(arctanh(x))) = sqrt((1+x)/(1-x)).

%F a(n) ~ n!*2*sqrt(2)*exp(Pi/2)/(exp(Pi)-1) * ((exp(Pi/2)+1)/(exp(Pi/2)-1))^n. - _Vaclav Kotesovec_, Oct 18 2013

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 81*x^4/4! + 605*x^5/5! + ...

%e where 1/A(tanh(x)) = cos(x) + sin(x).

%t CoefficientList[Series[1/(Sqrt[2]*Sin[Pi/4 + 1/2*Log[(1-x)/(1+x)]]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 18 2013 *)

%o (PARI) {a(n)=n!*polcoeff(1/(cos(atanh(x+x*O(x^n)))-sin(atanh(x+x*O(x^n)))), n)}

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 22 2011