|
%I #8 Jul 24 2013 22:34:00
%S 1,2,18,114,450,2298,10466,43314,184402,749490,2942274,11437026,
%T 43364818,161089130,589901682,2123791130,7531395154,26360805018,
%U 91057065522,310718196626,1048405959266,3499152601394,11559430074418,37818135048962,122582070331106,393830310786706,1254654362883954
%N O.g.f.: exp( Sum_{n>=1} (sigma(2*n^4) - sigma(n^4)) * x^n/n ).
%C Compare to the Jacobi theta_3 function:
%C 1 + 2*Sum_{n>=1} x^(n^2) = exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).
%C Here sigma(n) = A000203(n), the sum of the divisors of n.
%F O.g.f.: exp( Sum_{n>=1} A054785(n^4)*x^n/n ).
%F Logarithmic derivative equals A224903.
%F a(n) == 2 (mod 4) for n>0 (conjecture).
%e G.f.: A(x) = 1 + 2*x + 18*x^2 + 114*x^3 + 450*x^4 + 2298*x^5 +...
%e where
%e log(A(x)) = 2*x + 32*x^2/2 + 242*x^3/3 + 512*x^4/4 + 1562*x^5/5 +...+ A224903(n)*x^n/n +...
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^4)-sigma(m^4))*x^m/m)+x^2*O(x^n)), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A224903, A054785, A000203; variants: A195584, A215603, A225958.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 24 2013
|
