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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
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
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
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 29 2011
STATUS
approved