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.

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

Offset: 0

Author

Paul D. Hanna, Jul 29 2011

Keywords

• sign

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.

Examples

			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! +...
		

Links

• Eric Weisstein's World of Mathematics, Ramanujan Cos/Cosh Identity.

Crossrefs

Cf. A193540, A193541, A193543, A193544, A193545.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ JacobiDN[ x, -1]^2, {x, 0, n}]]; (* Michael Somos, Jun 17 2016 *)

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))}

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