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A193540 E.g.f.: Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L)/cosh(n*Pi)) where L = Lemniscate constant. 6
1, -1, 9, -153, 4977, -261009, 20039481, -2121958377, 296297348193, -52750142341281, 11662264481073129, -3134732109393169593, 1006734732695870345937, -380718482718134681818929, 167456229155543640166939161, -84761007600911799530893148937 (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 A193543.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

T. J. Stieltjes, LXV. Sur les dérivées de sec x, p. 181, Oeuvres complètes, tome 2, Noordhoff, 1918, 617 p.

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

FORMULA

Given e.g.f. A(x), define the e.g.f. of A193543:

B(x) = sqrt(2)*Pi/(2*L) * (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 A193541.

O.g.f. = 1/(1 + 1^2*x/(1 + 2*2^2*x/(1 + 3^2*x/(1 + 2*4^2*x/(1 + 5^2*x/(1 + 2*6^2*x/(1 + 7^2*x/(1 + 2*8^2*x/(1+...))))))))) (continued fraction).

G.f.: 1/Q(0) where p=2, Q(k) = 1 + x*(2*k+1)^2/( 1 + p*x*(2*k+2)^2/Q(k+1) ); (continued fraction due to T. J. Stieltjes). - Sergei N. Gladkovskii, Mar 22 2013

a(n) ~ (-1)^n * 2^(7*n + 4) * Pi^(n+1) * n^(2*n + 1/2) / (exp(2*n) * Gamma(1/4)^(4*n + 2)). - Vaclav Kotesovec, Nov 29 2020

EXAMPLE

E.g.f.: A(x) = 1 - x^2/2! + 9*x^4/4! - 153*x^6/6! + 4977*x^8/8! - 261009*x^10/10! + 20039481*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 A193543, 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! + 9*x^4/4! + 153*x^6/6! + 4977*x^8/8! + 261009*x^10/10! + 20039481*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 + 9*x^2 - 153*x^3 + 4977*x^4 - 261009*x^5 + 20039481*x^6 +...+ a(n)*x^n +...

O.g.f.: 1/(1 + x/(1 + 8*x/(1 + 9*x/(1 + 32*x/(1 + 25*x/(1 + 72*x/(1 + 49*x/(1 + 128*x/(1+...))))))))).

MATHEMATICA

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

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

Table[SeriesCoefficient[InverseSeries[Series[EllipticF[x, 1/2], {x, 0, 32}]], 2 n + 1] (2 n + 1)! 2^n, {n, 0, 15}] (* Benedict W. J. Irwin, Apr 04 2017 *)

Table[SeriesCoefficient[JacobiDN[Sqrt[2] x, 1/2], {x, 0, 2 k}] (2 k)!, {k, 0, 20}] (* Jan Mangaldan, Nov 28 2020 *)

nmax = 20; s = CoefficientList[Series[JacobiDN[Sqrt[2] x, 1/2], {x, 0, 2*nmax}], x] * Range[ 0, 2*nmax]!; Table[s[[2*n + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Nov 29 2020 *)

PROG

(PARI) {a(n)=local(R, L=2*(Pi/2)^(3/2)/gamma(3/4)^2);

R=Pi/(sqrt(2)*L)*(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. A193541, A193542, A193543, A193544, A193545.

Sequence in context: A045755 A009037 A012148 * A193543 A173982 A185759

Adjacent sequences: A193537 A193538 A193539 * A193541 A193542 A193543

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jul 29 2011

STATUS

approved

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Last modified December 6 19:33 EST 2022. Contains 358648 sequences. (Running on oeis4.)