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A193541 E.g.f.: sqrt(2)*L / (Pi*(1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi) )) where L = Lemniscate constant. 8
1, 1, -3, -27, 441, 11529, -442827, -23444883, 1636819569, 145703137041, -16106380394643, -2164638920874507, 347592265948756521, 65724760945840254489, -14454276753061349098587, -3658147171522531111996803, 1055646229815910768764248289 (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 A193544.

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 A193544:

B(x) = sqrt(2)*L / (Pi*(1 + 2*Sum_{n>=1} cosh(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 A193540.

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).

G.f.: 1/U(0) where U(k)= 1 - x*(2*k+1)^2/(1 + x*(2*k+2)^2/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jun 28 2012

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*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) +...)).

Let B(x) equal the e.g.f. of A193544, where:

B(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) +...))

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

a[ n_] := If[ n < 0, 0, With[{m = 2 n}, 2^n m! SeriesCoefficient[ JacobiND[ x, 1/2], {x, 0, m}]]]; (* Michael Somos, Oct 18 2011 *)

a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ JacobiDN[ x, -1], {x, 0, m}]]]; (* 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 +O(x^(2*n+1)))/cosh(m*Pi)));

round((2*n)!*polcoeff(R, 2*n))}

CROSSREFS

Cf. A159600, A193540, A193542, A193543, A193544, A193545.

Sequence in context: A279844 A159600 A159601 * A193544 A286306 A285239

Adjacent sequences: A193538 A193539 A193540 * A193542 A193543 A193544

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jul 29 2011

STATUS

approved

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Last modified December 9 02:22 EST 2022. Contains 358698 sequences. (Running on oeis4.)