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A193538 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^2/2 * x^n/n ). 0

%I #9 Mar 30 2012 18:37:28

%S 1,2,6,20,46,116,284,632,1414,3102,6536,13636,28020,56300,111888,

%T 219608,424694,813104,1540818,2888060,5366072,9884616,18050428,

%U 32713048,58851972,105113942,186505864,328821408,576153008,1003687444,1738735728,2995837872

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

%C 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

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

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

%F Self-convolution yields A177398.

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 46*x^4 + 116*x^5 + 284*x^6 +...

%e 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 +...

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

%Y Cf. A177398, A054785, A186690.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 29 2011

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