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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. 5
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.

REFERENCES

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

LINKS

Table of n, a(n) for n=0..15.

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

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 *)

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 August 25 06:07 EDT 2019. Contains 326323 sequences. (Running on oeis4.)