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A193538
O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^2/2 * x^n/n ).
0
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
0,2
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).
FORMULA
Self-convolution yields A177398.
EXAMPLE
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 +...
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^2/2*x^m/m)+x*O(x^n)), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 29 2011
STATUS
approved