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

1, 1, 41, 126, 1526, 5185, 46920, 176865, 1254608, 4986548, 30563031, 123868761, 683127011, 2793828323, 14223836013, 58127497582, 278433541834, 1130954381904, 5159127957638, 20767403083249, 91032595281699, 362455763000997, 1536849042738162

Offset: 0

Author

Paul D. Hanna, Jan 31 2012

Keywords

• nonn

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.

Examples

			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 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3200

Crossrefs

Cf. A156302, A178933, A000203 (sigma), A000041 (partitions), A361179.

Programs

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 *)

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)))

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)