A205572 - OEIS

%I #22 Sep 23 2013 09:05:21

%S 1,1,3,13,73,521,4443,44213,502993,6436561,91520883,1431459613,

%T 24424457113,451474855001,8987248462923,191682800678213,

%U 4360821252342433,105410131831623841,2697863748098734563,72885101748061044013,2072687894252786558953

%N E.g.f.: 1/(cos(x) - sinh(x)).

%C Radius of convergence of e.g.f. is |x| < r where r = 0.703290658863965... satisfies cos(r) = sinh(r).

%H Vincenzo Librandi, <a href="/A205572/b205572.txt">Table of n, a(n) for n = 0..200</a>

%F a(2^n + k) == a(k) (mod 2^n) for k>=0, n>=1 (conjecture).

%F E.g.f.: E(x) = 1/(cos(x) - sinh(x)) = 1/G(0) where G(k)= 1 -x/(4*k +1 - x*(4*k +1)/(4*k + 2 + x - 2*x*(2*k+1)/(4*k + 3 + x- x*(4*k+3)/(x -4*(k+1)/G(k+1))))); Radius of convergence of e.g.f.E(x)=1/G(0) is infinity; (continued fraction, 3rd kind, 5-step). - _Sergei N. Gladkovskii_, Jun 08 2012, Oct 03 2012

%F a(n) ~ n! * 2*exp(r)/((2*sin(r)*exp(r)+exp(2*r)+1)*r^(n+1)), where r = 0.7032906588639654... is defined in the comment. - _Vaclav Kotesovec_, Sep 22 2013

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 73*x^4/4! + 521*x^5/5! +...

%t CoefficientList[Series[1/(Cos[x]-Sinh[x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 22 2013 *)

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 29 2012