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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. 5
 1, 0, 2, 0, 0, 0, -144, 0, 0, 0, 96768, 0, 0, 0, -268240896, 0, 0, 0, 2111592333312, 0, 0, 0, -37975288540299264, 0, 0, 0, 1353569484565546795008, 0, 0, 0, -86498911610371173437669376, 0, 0, 0, 9198407234012051081051108278272, 0, 0, 0, -1536583522302562247445395779495133184 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429... Compare the definition with that of the dual sequence A193545. LINKS Eric Weisstein's World of Mathematics, Ramanujan Cos/Cosh Identity. FORMULA a(n) = -A193545(n) for n>=1. E.g.f.: dn(x, -1)^2 where dn() is a Jacobi elliptic function. - Michael Somos, Jun 17 2016 EXAMPLE 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! +... which equals the square of the e.g.f. B(x) of A193541: 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! +... MATHEMATICA a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiDN[ x, -1]^2, {x, 0, n}]]; (* Michael Somos, Jun 17 2016 *) PROG (PARI) {a(n)=local(R, L=2*(Pi/2)^(3/2)/gamma(3/4)^2); R=(sqrt(2)*L/Pi)/(1 + 2*suminf(m=1, cos(2*Pi*m*x/L +x*O(x^n))/cosh(m*Pi))); round(n!*polcoeff(R^2, n))} CROSSREFS Cf. A193540, A193541, A193543, A193544, A193545. Sequence in context: A057383 A218881 A169772 * A193545 A336399 A086260 Adjacent sequences: A193539 A193540 A193541 * A193543 A193544 A193545 KEYWORD sign AUTHOR Paul D. Hanna, Jul 29 2011 STATUS approved

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Last modified April 1 22:19 EDT 2023. Contains 361716 sequences. (Running on oeis4.)