A178933 - OEIS

%I #15 Oct 13 2012 14:50:55

%S 1,1,14,35,205,521,2507,6709,26712,73834,262431,724537,2384988,

%T 6552033,20289864,55244988,163342701,439201501,1251532060,3321188863,

%U 9177476977,24028568664,64709650590,167153761523,440300702427,1122562426240,2900254892900,7301575351055,18544013542057

%N Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

%C Compare with g.f. for partition numbers A000041: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

%C Similarly, exp( Sum_{n>=1} sigma(n)^2*x^n/n ) gives A156302.

%F a(0)=0 and a(n)=1/n*sum(k=1,n,sigma(k)^3*a(n-k)) for n>0.

%F G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^2 * x^(n*k) / n ). [_Paul D. Hanna_, Jan 31 2012]

%e G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...

%e such that, by definition,

%e log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 6^3*x^5/5 + 12^3*x^6/6 +...

%o (PARI) N=100;v=Vec(exp(sum(k=1,N,sigma(k)^3*x^k/k)+x*O(x^N)))

%o (PARI) a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^3*a(n-k)))

%o (PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*x^k/k)+x*O(x^n)), n)} /* _Paul D. Hanna_ */

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^2*x^(m*k)/m)+x*O(x^n))), n)} /* _Paul D. Hanna_ */

%Y Cf. A000203 (sigma), A000041 (partitions), A156302, A205797.

%K nonn

%O 0,3

%A _Joerg Arndt_, Dec 30 2010