cp's OEIS Frontend

A205797 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^4 * x^n/n ).

%I A205797 #14 Oct 30 2024 13:55:42
%S A205797 1,1,41,126,1526,5185,46920,176865,1254608,4986548,30563031,123868761,
%T A205797 683127011,2793828323,14223836013,58127497582,278433541834,
%U A205797 1130954381904,5159127957638,20767403083249,91032595281699,362455763000997,1536849042738162
%N A205797 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^4 * x^n/n ).
%C A205797 Compare with g.f. for partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.
%H A205797 Vaclav Kotesovec, <a href="/A205797/b205797.txt">Table of n, a(n) for n = 0..3200</a>
%F A205797 a(n) = (1/n)*Sum_{k=1..n} sigma(k)^4*a(n-k) for n>0, with a(0) = 1.
%F A205797 G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^3 * x^(n*k) / n ).
%F A205797 From _Vaclav Kotesovec_, Oct 30 2024: (Start)
%F A205797 log(a(n)) ~ 5^(4/5) * c^(1/5) * Pi^(6/5) * zeta(3)^(1/5) * zeta(5)^(1/5) * n^(4/5) / (2^(9/5) * 3^(2/5)), where c = Product_{primes p} (1 + 3/p^2 + 5/p^3 + 3/p^4 + 3/p^5 + 5/p^6 + 3/p^7 + 1/p^9) = 6.04468280906514379869287397833397910321972833863778...
%F A205797 Equivalently, log(a(n)) ~ 3.967005157823944635858584839447899089435134... * n^(4/5). (End)
%e A205797 G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 5185*x^5 +...
%e A205797 such that, by definition,
%e A205797 log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 6^4*x^5/5 + 12^4*x^6/6 +...
%t A205797 nmax = 30; $RecursionLimit -> Infinity; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[1, k]^4 * a[n-k], {k, 1, n}]/n]; Table[a[n], {n, 0, nmax}] (* _Vaclav Kotesovec_, Oct 30 2024 *)
%o A205797 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^4*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */
%o A205797 (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sum(k=1, n\m, sigma(m*k)^3*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */
%o A205797 (PARI) a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^4*a(n-k)))
%Y A205797 Cf. A156302, A178933, A000203 (sigma), A000041 (partitions), A361179.
%K A205797 nonn
%O A205797 0,3
%A A205797 _Paul D. Hanna_, Jan 31 2012