A196354 E.g.f.: 1 + Sum_{n>=1} 2*cosh(n*x) * x^(n^2).

1, 2, 0, 6, 48, 10, 2880, 14, 53760, 725778, 645120, 359251222, 6082560, 42032390426, 49201152, 2648040595230, 41845937602560, 115757203161634, 102437981698129920, 3958896348126758, 51901909523009372160, 113368395423628842, 12788630502806158049280

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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.

E.g.f.: theta_3( i q/2, q ). - Michael Somos, Oct 29 2011

Example

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! +...
The e.g.f. A(x) may be expressed by the series:
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 +...
and by Jacobi's triple product:
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)) *...

Mathematica

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 *)

Prog

(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)}
(PARI) /* By Jacobi's Triple Product Identity: */
{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)}

Crossrefs

Cf. A190791, A197707, A192502.

Sequence in context: A280217 A079203 A151751 * A305620 A294470 A283654

Adjacent sequences: A196351 A196352 A196353 * A196355 A196356 A196357

Keyword

nonn,look

Author

Paul D. Hanna, Oct 28 2011

Status

approved