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A177399 O.g.f.: exp( Sum_{n>=1} (sigma(2n)-sigma(n))^n * x^n/n ). 3

%I #2 Mar 30 2012 18:37:21

%S 1,2,10,188,1414,53596,2923652,44668152,651967302,605335444140,

%T 7564881098284,157357140966472,96537385644719004,695895399853879448,

%U 86358988630956719304,1103071610291574716763120

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

%C Here sigma(n) = A000203(n) is the sum of divisors of n.

%C Compare g.f. to the formula for Jacobi theta_4(x) given by:

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

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

%e G.f.: A(x) = 1 + 2*x + 10*x^2 + 188*x^3 + 1414*x^4 + 53596*x^5 +...

%e log(A(x)) = 2*x + 4^2*x^2/2 + 8^3*x^3/3 + 8^4*x^4/4 + 12^5*x^5/5 +...+ A054785(n)^n*x^n/n +...

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

%Y Cf. A054785, A000203, A177398, A155200.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 30 2010

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