cp's OEIS Frontend

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

%I A193539 #9 Mar 30 2012 18:37:28
%S A193539 1,8,64,512,3200,19392,112128,598016,3088896,15362408,73331264,
%T A193539 340653056,1538392064,6762336448,29072665600,122299068416,
%U A193539 504128374784,2040557142592,8116582974656,31760991869952,122408808197120,464983163273216,1742277357389312
%N A193539 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)-sigma(n))^3 * x^n/n ).
%C A193539 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 A193539 _ theta_4(x) = exp( Sum_{n>=1} (sigma(n)-sigma(2*n))*x^n/n )
%C A193539 where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).
%e A193539 G.f.: A(x) = 1 + 8*x + 64*x^2 + 512*x^3 + 3200*x^4 + 19392*x^5 +...
%e A193539 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 +...
%o A193539 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m)-sigma(m))^3*x^m/m)+x*O(x^n)), n)}
%Y A193539 Cf. A177398, A054785, A186690.
%K A193539 nonn
%O A193539 0,2
%A A193539 _Paul D. Hanna_, Jul 30 2011