A224902 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^4) - sigma(n^4)) * x^n/n ).

1, 2, 18, 114, 450, 2298, 10466, 43314, 184402, 749490, 2942274, 11437026, 43364818, 161089130, 589901682, 2123791130, 7531395154, 26360805018, 91057065522, 310718196626, 1048405959266, 3499152601394, 11559430074418, 37818135048962, 122582070331106, 393830310786706, 1254654362883954

Offset: 0

Author

Paul D. Hanna, Jul 24 2013

Keywords

• nonn

Comments

Compare to the Jacobi theta_3 function:

1 + 2*Sum_{n>=1} x^(n^2) = exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).

Here sigma(n) = A000203(n), the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 114*x^3 + 450*x^4 + 2298*x^5 +...
where
log(A(x)) = 2*x + 32*x^2/2 + 242*x^3/3 + 512*x^4/4 + 1562*x^5/5 +...+ A224903(n)*x^n/n +...
		

Crossrefs

Cf. A224903, A054785, A000203; variants: A195584, A215603, A225958.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^4)-sigma(m^4))*x^m/m)+x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

O.g.f.: exp( Sum_{n>=1} A054785(n^4)*x^n/n ).

Logarithmic derivative equals A224903.

a(n) == 2 (mod 4) for n>0 (conjecture).