A193539 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^3 * x^n/n ).

1, 8, 64, 512, 3200, 19392, 112128, 598016, 3088896, 15362408, 73331264, 340653056, 1538392064, 6762336448, 29072665600, 122299068416, 504128374784, 2040557142592, 8116582974656, 31760991869952, 122408808197120, 464983163273216, 1742277357389312

Offset: 0

Author

Paul D. Hanna, Jul 30 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 + 8*x + 64*x^2 + 512*x^3 + 3200*x^4 + 19392*x^5 +...
log(A(x)) = 2^3*x + 4^3*x^2/2 + 8^3*x^3/3 + 8^3*x^4/4 + 12^3*x^5/5 + 16^3*x^6/6 + 16^3*x^7/7 + 16^3*x^8/8 + 26^3*x^9/9 +...+ A054785(n)^3*x^n/n +...
		

Crossrefs

Cf. A177398, A054785, A186690.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^3*x^m/m)+x*O(x^n)), n)}