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A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).

%I A178933 #21 Oct 30 2024 12:56:10
%S A178933 1,1,14,35,205,521,2507,6709,26712,73834,262431,724537,2384988,
%T A178933 6552033,20289864,55244988,163342701,439201501,1251532060,3321188863,
%U A178933 9177476977,24028568664,64709650590,167153761523,440300702427,1122562426240,2900254892900,7301575351055,18544013542057
%N A178933 Generating function exp( sum(n>=1, sigma(n)^3*x^n/n ) ).
%C A178933 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 A178933 Similarly, exp( Sum_{n>=1} sigma(n)^2*x^n/n ) gives A156302.
%H A178933 Vaclav Kotesovec, <a href="/A178933/b178933.txt">Table of n, a(n) for n = 0..6700</a>
%F A178933 a(0)=0 and a(n)=1/n*sum(k=1,n,sigma(k)^3*a(n-k)) for n>0.
%F A178933 G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^2 * x^(n*k) / n ). [_Paul D. Hanna_, Jan 31 2012]
%F A178933 From _Vaclav Kotesovec_, Oct 30 2024: (Start)
%F A178933 log(a(n)) ~ 2^(7/4) * c^(1/4) * Pi^(3/2) * zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)), where c = Product_{primes p} (1 + 2/p^2 + 2/p^3 + 1/p^5) = 2.8359835743341928677044245715803848964089818378791769798895797934086403174189...
%F A178933 Equivalently, log(a(n)) ~ 3.2753680082113515869730831738879060384726246... * n^(3/4). (End)
%e A178933 G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...
%e A178933 such that, by definition,
%e A178933 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 +...
%t A178933 nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1,k]^3 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 30 2024 *)
%o A178933 (PARI) N=100;v=Vec(exp(sum(k=1,N,sigma(k)^3*x^k/k)+x*O(x^N)))
%o A178933 (PARI) a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^3*a(n-k)))
%o A178933 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*x^k/k)+x*O(x^n)), n)} /* _Paul D. Hanna_ */
%o A178933 (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 A178933 Cf. A000203 (sigma), A000041 (partitions), A156302, A205797, A361147.
%K A178933 nonn
%O A178933 0,3
%A A178933 _Joerg Arndt_, Dec 30 2010