A193541 - OEIS

%I #31 Sep 29 2017 04:20:53

%S 1,1,-3,-27,441,11529,-442827,-23444883,1636819569,145703137041,

%T -16106380394643,-2164638920874507,347592265948756521,

%U 65724760945840254489,-14454276753061349098587,-3658147171522531111996803,1055646229815910768764248289

%N E.g.f.: sqrt(2)*L / (Pi*(1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi) )) where L = Lemniscate constant.

%C L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...

%C Compare the definition with that of the dual sequence A193544.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanCosCoshIdentity.html">Ramanujan Cos/Cosh Identity</a>.

%F Given e.g.f. A(x), define the e.g.f. B(x) of A193544:

%F B(x) = sqrt(2)*L / (Pi*(1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi) )),

%F then A(x)^2 + B(x)^2 = 2 by Ramanujan's cos/cosh identity.

%F E.g.f. equals the reciprocal of the e.g.f. of A193540.

%F O.g.f.: 1/(1 - 1^2*x/(1 + 2^2*x/(1 - 3^2*x/(1 + 4^2*x/(1 - 5^2*x/(1 + 6^2*x/(1 - 7^2*x/(1 + 8^2*x/(1-...))))))))) (continued fraction).

%F G.f.: 1/U(0) where U(k)= 1 - x*(2*k+1)^2/(1 + x*(2*k+2)^2/U(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Jun 28 2012

%F G.f.:  Q(0), where Q(k) = 1 - x*(2*k+1)^2/(x*(2*k+1)^2 - 1/(1 - x*(2*k+2)^2/(x*(2*k+2)^2 + 1/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 21 2013

%e E.g.f.: A(x) = 1 + x^2/2! - 3*x^4/4! - 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! - 442827*x^12/12! +...+ a(n)*x^(2*n)/(2*n)! +...

%e where

%e A(x) = sqrt(2)*L/(Pi*(1 + 2*cos(2*Pi*x/L)/cosh(Pi) + 2*cos(4*Pi*x/L)/cosh(2*Pi) + 2*cos(6*Pi*x/L)/cosh(3*Pi) +...)).

%e Let B(x) equal the e.g.f. of A193544, where:

%e B(x) = sqrt(2)*L/(Pi*(1 + 2*cosh(2*Pi*x/L)/cosh(Pi) + 2*cosh(4*Pi*x/L)/cosh(2*Pi) + 2*cosh(6*Pi*x/L)/cosh(3*Pi) +...))

%e explicitly,

%e B(x) = 1 - x^2/2! - 3*x^4/4! + 27*x^6/6! + 441*x^8/8! - 11529*x^10/10! - 442827*x^12/12! +...

%e then A(x)^2 + B(x)^2 = 2

%e as illustrated by:

%e A(x)^2 = 1 + 2*x^2/2! - 144*x^6/6! + 96768*x^10/10! - 268240896*x^14/14! +...

%e B(x)^2 = 1 - 2*x^2/2! + 144*x^6/6! - 96768*x^10/10! + 268240896*x^14/14! +...

%e ...

%e O.g.f.: 1 + x - 3*x^2 - 27*x^3 + 441*x^4 + 11529*x^5 - 442827*x^6 +...+ a(n)*x^n +...

%e O.g.f.: 1/(1 - x/(1 + 4*x/(1 - 9*x/(1 + 16*x/(1 - 25*x/(1 + 36*x/(1 - 49*x/(1 + 64*x/(1-...))))))))).

%t a[ n_] := If[ n < 0, 0, With[{m = 2 n}, 2^n m! SeriesCoefficient[ JacobiND[ x, 1/2], {x, 0, m}]]]; (* _Michael Somos_, Oct 18 2011 *)

%t a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ JacobiDN[ x, -1], {x, 0, m}]]]; (* _Michael Somos_, Jun 17 2016 *)

%o (PARI) {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);

%o R=(sqrt(2)*L/Pi)/(1 + 2*suminf(m=1,cos(2*Pi*m*x/L +O(x^(2*n+1)))/cosh(m*Pi)));

%o round((2*n)!*polcoeff(R,2*n))}

%Y Cf. A159600, A193540, A193542, A193543, A193544, A193545.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jul 29 2011