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A198296 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n) ).

%I A198296 #17 Mar 30 2012 18:37:31
%S A198296 1,1,2,3,6,8,17,22,44,62,115,154,311,409,754,1070,1949,2639,4917,6645,
%T A198296 12055,16916,29594,40719,73907,100959,176010,248207,429626,594220,
%U A198296 1040624,1436936,2473555,3486360,5901887,8233872,14174779,19689223,33203829,46967767
%N A198296 G.f.: exp( Sum_{n>=1} (x^n/n) / Product_{d|n} (1 - d*x^n) ).
%C A198296 Logarithmic derivative yields A198299.
%H A198296 Paul D. Hanna, <a href="/A198296/b198296.txt">Table of n, a(n) for n = 0..500</a>
%F A198296 G.f.: exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k ) ), where sigma(n,k) is the sum of the k-th powers of the divisors of n.
%e A198296 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 17*x^6 + 22*x^7 +...
%e A198296 such that, by definition:
%e A198296 log(A(x)) = x/(1-x) + (x^2/2)/((1-x^2)*(1-2*x^2)) + (x^3/3)/((1-x^3)*(1-3*x^3)) + (x^4/4)/((1-x^4)*(1-2*x^4)*(1-4*x^4)) + (x^5/5)/((1-x^5)*(1-5*x^5)) + (x^6/6)/((1-x^6)*(1-2*x^6)*(1-3*x^6)*(1-6*x^6)) +...+ (x^n/n)/Product_{d|n} (1-d*x^n) +...
%e A198296 Also, we have the identity:
%e A198296 log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 +...)*x
%e A198296 + (1 + 3*x^2 + 7*x^4 + 15*x^6 + 31*x^8 +...)*x^2/2
%e A198296 + (1 + 4*x^3 + 13*x^6 + 40*x^9 + 121*x^12 +...)*x^3/3
%e A198296 + (1 + 7*x^4 + 35*x^8 + 155*x^12 + 651*x^16 +...)*x^4/4
%e A198296 + (1 + 6*x^5 + 31*x^10 + 156*x^15 + 781*x^20 +...)*x^5/5
%e A198296 + (1 + 12*x^6 + 97*x^12 + 672*x^18 + 4333*x^24 +...)*x^6/6 +...
%e A198296 + exp( Sum_{k>=1} sigma(n,k)*x^(n*k)/k )*x^n/n +...
%e A198296 Explicitly, the logarithm begins:
%e A198296 log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 11*x^4/4 + 6*x^5/5 + 36*x^6/6 + 8*x^7/7 + 83*x^8/8 + 49*x^9/9 + 178*x^10/10 +...+ A198299(n)*x^n/n +...
%o A198296 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*exp(sum(k=1,n\m,sigma(m,k)*x^(m*k)/k)+x*O(x^n)))),n)}
%o A198296 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*exp(sumdiv(m,d,-log(1-d*x^m+x*O(x^n)))))),n)}
%Y A198296 Cf. A198299 (log), A203320, A185301, A198304.
%K A198296 nonn
%O A198296 0,3
%A A198296 _Paul D. Hanna_, Jan 26 2012