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%I #31 Nov 29 2020 03:53:49
%S 1,1,9,153,4977,261009,20039481,2121958377,296297348193,
%T 52750142341281,11662264481073129,3134732109393169593,
%U 1006734732695870345937,380718482718134681818929,167456229155543640166939161,84761007600911799530893148937,48919649166315485705652984573633
%N E.g.f.: Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi)) where L = Lemniscate constant.
%C L = Lemniscate constant = 2*(Pi/2)^(3/2)/gamma(3/4)^2 = 2.62205755429...
%C Compare the definition with that of the dual sequence A193540.
%H Vaclav Kotesovec, <a href="/A193543/b193543.txt">Table of n, a(n) for n = 0..234</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanCosCoshIdentity.html">Ramanujan Cos/Cosh Identity</a>.
%F E.g.f.: cosh( Series_Reversion( Integral 1/sqrt( cosh(2*x) ) dx ) ). - _Paul D. Hanna_, Aug 14 2017
%F E.g.f.: sqrt(1 + S(x)^2), where S(x) is the e.g.f. of A289695. - _Paul D. Hanna_, Aug 14 2017
%F E.g.f.: 1 + Integral S(x) * sqrt(1 + 2*S(x)^2) dx, where S(x) is the e.g.f. of A289695. - _Paul D. Hanna_, Aug 14 2017
%F ...
%F Given e.g.f. A(x), define the e.g.f. of A193540:
%F B(x) = Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} cos(2*Pi*n*x/L) / cosh(n*Pi)),
%F then A(x)^-2 + B(x)^-2 = 2 by Ramanujan's cos/cosh identity.
%F ...
%F E.g.f. equals the reciprocal of the e.g.f. of A193544.
%F ...
%F 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).
%F O.g.f.: Pi/(sqrt(2)*L) * (1 + 2*Sum_{n>=1} 1/(1 - (2*n*Pi/L)^2*x) / cosh(n*Pi)) where L = Lemniscate constant. - _Paul D. Hanna_, Aug 29 2012
%F ...
%F a(n) = sqrt(2)*Pi/L * Sum_{k>=1} (2*k*Pi/L)^(2*n) / cosh(k*Pi) for n>0 where L = Lemniscate constant. - _Paul D. Hanna_, Aug 29 2012
%F ...
%F G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)^2/(x*(2*k+1)^2 - 1/(1 - 2*x*(2*k+2)^2/(2*x*(2*k+2)^2 - 1/Q(k+1) ))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 21 2013
%F a(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
%e 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)! +...
%e where
%e 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) +...
%e Let B(x) equal the e.g.f. of A193540, where:
%e 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) +...
%e explicitly,
%e 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! +...
%e then A(x)^-2 + B(x)^-2 = 2
%e as illustrated by:
%e A(x)^-2 = 1 - 2*x^2/2! + 144*x^6/6! - 96768*x^10/10! + 268240896*x^14/14! +...
%e B(x)^-2 = 1 + 2*x^2/2! - 144*x^6/6! + 96768*x^10/10! - 268240896*x^14/14! +...
%e ...
%e O.g.f.: 1 + x + 9*x^2 + 153*x^3 + 4977*x^4 + 261009*x^5 + 20039481*x^6 +...+ a(n)*x^n +...
%e 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-...))))))))).
%t nmax = 20; s = CoefficientList[Series[JacobiDN[Sqrt[2]*x, 1/2], {x, 0, 2*nmax}], x] * Range[ 0, 2*nmax]!; Table[(-1)^n * s[[2*n + 1]], {n, 0, nmax}] (* _Vaclav Kotesovec_, Nov 29 2020 *)
%o (PARI) {a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2);if(n==0,1,sqrt(2)*Pi/L*suminf(k=1,(2*k*Pi/L)^(2*n)/cosh(k*Pi)))} \\ _Paul D. Hanna_, Aug 29 2012
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
%o R=Pi/(sqrt(2)*L)*(1 + 2*suminf(m=1,cosh(2*Pi*m*x/L +O(x^(2*n+1)))/cosh(m*Pi)));
%o round((2*n)!*polcoeff(R,2*n))}
%o (PARI) {a(n)=local(R,L=2*(Pi/2)^(3/2)/gamma(3/4)^2);
%o R=Pi/(sqrt(2)*L)*(1 + 2*suminf(m=1,1/(1 - (2*m*Pi/L)^2*x+x*O(x^n))/cosh(m*Pi)));
%o round(polcoeff(R,n))} \\ _Paul D. Hanna_, Aug 29 2012
%o (PARI) {a(n) = my(C=1); C = cosh( serreverse( intformal( 1/sqrt( cosh(2*x +O(x^(2*n+1))) ) ) ) ); (2*n)!*polcoeff(C,2*n)}
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Aug 14 2017
%Y Cf. A193540, A193541, A193542, A193544, A193545.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 29 2011
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