A196354 - OEIS

%I #28 Sep 03 2017 05:13:11

%S 1,2,0,6,48,10,2880,14,53760,725778,645120,359251222,6082560,

%T 42032390426,49201152,2648040595230,41845937602560,115757203161634,

%U 102437981698129920,3958896348126758,51901909523009372160,113368395423628842,12788630502806158049280

%N E.g.f.: 1 + Sum_{n>=1} 2*cosh(n*x) * x^(n^2).

%H Paul D. Hanna, <a href="/A196354/b196354.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: Product_{n>=1} (1 - x^(2*n))*(1 + x^(2*n-1)*exp(x))*(1 + x^(2*n-1)/exp(x)), due to the Jacobi triple product identity.

%F E.g.f.: theta_3( i q/2, q ). - _Michael Somos_, Oct 29 2011

%e E.g.f.: A(x) = 1 + 2*x + 0*x^2/2! + 6*x^3/3! + 48*x^4/4! + 10*x^5/5! + 2880*x^6/6! + 14*x^7/7! + 53760*x^8/8! +...

%e The e.g.f. A(x) may be expressed by the series:

%e A(x) = 1 + 2*cosh(x)*x + 2*cosh(2*x)*x^4 + 2*cosh(3*x)*x^9 + 2*cosh(4*x)*x^16 + 2*cosh(5*x)*x^25 +...

%e and by Jacobi's triple product:

%e A(x) = (1-x^2)*(1+x*exp(x))*(1+x/exp(x)) * (1-x^4)*(1+x^3*exp(x))*(1+x^3/exp(x)) * (1-x^6)*(1+x^5*exp(x))*(1+x^5/exp(x)) * (1-x^8)*(1+x^7*exp(x))*(1+x^7/exp(x)) *...

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (Series[ EllipticTheta[ 3, x I/2, q], {q, 0, n}] // Normal // TrigToExp) /. {x -> q}, {q, 0, n}]] (* _Michael Somos_, Nov 18 2011 *)

%o (PARI) {a(n)=local(A=1+x); A=1+sum(m=1,sqrtint(n+1), 2*cosh(m*x+x*O(x^n))*x^(m^2)); n!*polcoeff(A, n)}

%o (PARI) /* By Jacobi's Triple Product Identity: */

%o {a(n)=local(A=1+x); A=prod(m=1, n\2+1, (1-x^(2*m))*(1+exp(x+x*O(x^n))*x^(2*m-1))*(1+exp(-x+x*O(x^n))*x^(2*m-1)+x*O(x^n))); n!*polcoeff(A, n)}

%Y Cf. A190791, A197707, A192502.

%K nonn,look

%O 0,2

%A _Paul D. Hanna_, Oct 28 2011