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A193542 E.g.f.: 2*L^2/(Pi^2*(1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi) )^2) where L = Lemniscate constant.

%I A193542 #8 Feb 16 2025 08:33:15
%S A193542 1,0,2,0,0,0,-144,0,0,0,96768,0,0,0,-268240896,0,0,0,2111592333312,0,
%T A193542 0,0,-37975288540299264,0,0,0,1353569484565546795008,0,0,0,
%U A193542 -86498911610371173437669376,0,0,0,9198407234012051081051108278272,0,0,0,-1536583522302562247445395779495133184
%N A193542 E.g.f.: 2*L^2/(Pi^2*(1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi) )^2) where L = Lemniscate constant.
%C A193542 L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...
%C A193542 Compare the definition with that of the dual sequence A193545.
%H A193542 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanCosCoshIdentity.html">Ramanujan Cos/Cosh Identity</a>.
%F A193542 a(n) = -A193545(n) for n>=1.
%F A193542 E.g.f.: dn(x, -1)^2 where dn() is a Jacobi elliptic function. - _Michael Somos_, Jun 17 2016
%e A193542 E.g.f.: A(x) = 1 + 2*x^2/2! - 144*x^6/6! + 96768*x^10/10! - 268240896*x^14/14! +...+ a(n)*x^n/n! +...
%e A193542 which equals the square of the e.g.f. B(x) of A193541:
%e A193542 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! +...
%t A193542 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiDN[ x, -1]^2, {x, 0, n}]]; (* _Michael Somos_, Jun 17 2016 *)
%o A193542 (PARI) {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
%o A193542 R=(sqrt(2)*L/Pi)/(1 + 2*suminf(m=1,cos(2*Pi*m*x/L +x*O(x^n))/cosh(m*Pi)));
%o A193542 round(n!*polcoeff(R^2,n))}
%Y A193542 Cf. A193540, A193541, A193543, A193544, A193545.
%K A193542 sign
%O A193542 0,3
%A A193542 _Paul D. Hanna_, Jul 29 2011