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

1, 1, 14, 35, 205, 521, 2507, 6709, 26712, 73834, 262431, 724537, 2384988, 6552033, 20289864, 55244988, 163342701, 439201501, 1251532060, 3321188863, 9177476977, 24028568664, 64709650590, 167153761523, 440300702427, 1122562426240, 2900254892900, 7301575351055, 18544013542057

Offset: 0

Author

Joerg Arndt, Dec 30 2010

Keywords

• nonn

Comments

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.

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

Examples

			G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 521*x^5 + 2507*x^6 +...
such that, by definition,
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 +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..6700

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A156302, A205797, A361147.

Programs

Mathematica

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 *)

PARI

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

PARI

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

PARI

{a(n)=polcoeff(exp(sum(k=1, n, sigma(k)^3*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)^2*x^(m*k)/m)+x*O(x^n))), n)} /* Paul D. Hanna */

Formula

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

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

From Vaclav Kotesovec, Oct 30 2024: (Start)

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...

Equivalently, log(a(n)) ~ 3.2753680082113515869730831738879060384726246... * n^(3/4). (End)