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A193544 E.g.f.: sqrt(2)*(L/Pi) / (1 + 2*Sum_{n>=1} cosh(2*Pi*n*x/L)/cosh(n*Pi)) where L = Lemniscate constant. 10

%I #30 Sep 29 2017 04:54:12

%S 1,-1,-3,27,441,-11529,-442827,23444883,1636819569,-145703137041,

%T -16106380394643,2164638920874507,347592265948756521,

%U -65724760945840254489,-14454276753061349098587,3658147171522531111996803,1055646229815910768764248289

%N E.g.f.: sqrt(2)*(L/Pi) / (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 A193541.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanCosCoshIdentity.html">Ramanujan Cos/Cosh Identity</a>.

%F Given e.g.f. A(x), define the e.g.f. B(x) of A193541:

%F B(x) = sqrt(2)*L / (Pi*(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 A193543.

%F ...

%F 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).

%F O.g.f.: (Pi/L) * (1 + 2*Sum_{n>=1} (-1)^n/(1 - (2*n*Pi/L)^2*x) / cosh(n*Pi)) where L = Lemniscate constant. - _Paul D. Hanna_, Aug 29 2012

%F ...

%F a(n) = 2*Pi/L * Sum_{k>=1} (-1)^k*(2*k*Pi/L)^(2*n) / cosh(k*Pi) for n>0 where L = Lemniscate constant. - _Paul D. Hanna_, Aug 29 2012

%F G.f.: 1/Q(0), where Q(k)= 1 + x*(2*k+1)^2/(1 - x*(2*k+2)^2/Q(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Apr 27 2013

%F 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

%e 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)! +...

%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 A193541, 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! - 3*x^4/4! - 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! - 442827*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 - 3*x^2 + 27*x^3 + 441*x^4 - 11529*x^5 - 442827*x^6 +...+ a(n)*x^n +...

%e 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+...))))))))).

%t L = 2*(Pi/2)^(3/2)/Gamma[3/4]^2; a[0] = 1; a[n_] := 2*Pi/L*NSum[(-1)^k * (2*k*Pi/L)^(2*n)/Cosh[k*Pi], {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 50] // Round; Table[a[n], {n, 0, 16}] (* _Jean-Fran├žois Alcover_, Sep 29 2017 *)

%o (PARI) {a(n)=local(L=2*(Pi/2)^(3/2)/gamma(3/4)^2); if(n==0, 1, 2*Pi/L*suminf(k=1, (-1)^k*(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=(sqrt(2)*L/Pi)/(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/L)*(1 + 2*suminf(m=1,(-1)^m/(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

%Y Cf. A159600, A193540, A193541, A193542, A193543, A193545.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jul 29 2011

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