A193538 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^2/2 * x^n/n ).

1, 2, 6, 20, 46, 116, 284, 632, 1414, 3102, 6536, 13636, 28020, 56300, 111888, 219608, 424694, 813104, 1540818, 2888060, 5366072, 9884616, 18050428, 32713048, 58851972, 105113942, 186505864, 328821408, 576153008, 1003687444, 1738735728, 2995837872

Offset: 0

Author

Paul D. Hanna, Jul 29 2011

Keywords

• nonn

Comments

Here sigma(n) = A000203(n) is the sum of divisors of n. Compare g.f. to the formula for Jacobi theta_4(x) given by

theta_4(x) = exp( Sum_{n>=1} (sigma(n)-sigma(2*n))*x^n/n )

where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 46*x^4 + 116*x^5 + 284*x^6 +...
log(A(x)) = 2^2*x/2 + 4^2*x^2/4 + 8^2*x^3/6 + 8^2*x^4/8 + 12^2*x^5/10 + 16^2*x^6/12 + 16^2*x^7/14 + 16^2*x^8/16 + 26^2*x^9/18 +...+ A054785(n)^2/2*x^n/n +...
		

Crossrefs

Cf. A177398, A054785, A186690.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^2/2*x^m/m)+x*O(x^n)), n)}

Formula

Self-convolution yields A177398.