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A193545 E.g.f.: 2*L^2/(Pi^2*(1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi) )^2) where L = Lemniscate constant.

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