A193544 E.g.f.: sqrt(2)*(L/Pi) / (1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi)) where L = Lemniscate constant.

1, -1, -3, 27, 441, -11529, -442827, 23444883, 1636819569, -145703137041, -16106380394643, 2164638920874507, 347592265948756521, -65724760945840254489, -14454276753061349098587, 3658147171522531111996803, 1055646229815910768764248289

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 A193541.

Links

Table of n, a(n) for n=0..16.

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

Formula

Given e.g.f. A(x), define the e.g.f. B(x) of A193541:

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

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

...

E.g.f. equals the reciprocal of the e.g.f. of A193543.

...

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).

O.g.f.: (Pi/L) * (1 + 2*Sum_{n>=1} (-1)^n/(1 - (2*n*Pi/L)^2*x) / cosh(n*Pi)) where L = Lemniscate constant. - Paul D. Hanna, Aug 29 2012

...

a(n) = 2*Pi/L * Sum_{k>=1} (-1)^k*(2*k*Pi/L)^(2*n) / cosh(k*Pi) for n>0 where L = Lemniscate constant. - Paul D. Hanna, Aug 29 2012

G.f.: 1/Q(0), where Q(k)= 1 + x*(2*k+1)^2/(1 - x*(2*k+2)^2/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 27 2013

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

Example

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)! +...
where
A(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) +...)).
Let B(x) equal the e.g.f. of A193541, where:
B(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) +...))
explicitly,
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! +...
then A(x)^2 + B(x)^2 = 2
as illustrated by:
A(x)^2 = 1 - 2*x^2/2! + 144*x^6/6! - 96768*x^10/10! + 268240896*x^14/14! +...
B(x)^2 = 1 + 2*x^2/2! - 144*x^6/6! + 96768*x^10/10! - 268240896*x^14/14! +...
...
O.g.f.: 1 - x - 3*x^2 + 27*x^3 + 441*x^4 - 11529*x^5 - 442827*x^6 +...+ a(n)*x^n +...
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+...))))))))).

Mathematica

L = 2*(Pi/2)^(3/2)/Gamma[3/4]^2; a[0] = 1; a[n_] := 2*Pi/L*NSum[(-1)^k * (2*k*Pi/L)^(2*n)/Cosh[k*Pi], {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 50] // Round; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Sep 29 2017 *)

Prog

(PARI) {a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2); if(n==0, 1, 2*Pi/L*suminf(k=1, (-1)^k*(2*k*Pi/L)^(2*n)/cosh(k*Pi)))} \\ Paul D. Hanna, Aug 29 2012
for(n=0, 20, print1(a(n), ", "))
(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, cosh(2*Pi*m*x/L +O(x^(2*n+1)))/cosh(m*Pi)));
round((2*n)!*polcoeff(R, 2*n))}
(PARI) {a(n)=local(R, L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
R=(Pi/L)*(1 + 2*suminf(m=1, (-1)^m/(1 - (2*m*Pi/L)^2*x+x*O(x^n))/cosh(m*Pi)));
round(polcoeff(R, n))} \\ Paul D. Hanna, Aug 29 2012

Crossrefs

Cf. A159600, A193540, A193541, A193542, A193543, A193545.

Sequence in context: A159600 A159601 A193541 * A286306 A285239 A111844

Adjacent sequences: A193541 A193542 A193543 * A193545 A193546 A193547

Keyword

sign

Author

Paul D. Hanna, Jul 29 2011

Status

approved