OFFSET
0,3
COMMENTS
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.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..3200
FORMULA
a(n) = (1/n)*Sum_{k=1..n} sigma(k)^4*a(n-k) for n>0, with a(0) = 1.
G.f.: exp( Sum_{n>=1} Sum_{k>=1} sigma(n*k)^3 * x^(n*k) / n ).
From Vaclav Kotesovec, Oct 30 2024: (Start)
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...
Equivalently, log(a(n)) ~ 3.967005157823944635858584839447899089435134... * n^(4/5). (End)
EXAMPLE
G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 5185*x^5 +...
such that, by definition,
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 +...
MATHEMATICA
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 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^4*x^k/k)+x*O(x^n)), n)} /* Paul D. Hanna */
(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 */
(PARI) a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^4*a(n-k)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 31 2012
STATUS
approved