A028342 Expansion of Product_{i>=1} (1 - x^i)^(-1/i); also of exp(Sum_{n>=1} (d(n)*x^n/n)) where d is number of divisors function.

1, 1, 3, 11, 59, 339, 2629, 20677, 202089, 2066201, 24322931, 296746251, 4193572723, 59806188571, 954679763829, 15845349818789, 285841314451409, 5293203821406897, 106976406006818659, 2201383054398314251

Offset: 0

Author

Jeffrey Shallit

Keywords

• nonn
• easy

Comments

From Peter Bala, Nov 14 2017: (Start)

It appears that the sequence taken modulo 10 is periodic with period 10. More generally, we conjecture

(1) for k odd, a(n+k) + a(n) is divisible by k: if true, then for k odd, the sequence a(n) taken modulo k would be periodic with period dividing 2*k.

(2) for even k congruent to 0, 2 or 6 modulo 8 then a(n+k) - a(n) is divisible by k; in these cases the sequence a(n) taken modulo k would be periodic with period dividing k.

(3) for even k congruent to 4 modulo 8 then 2*( a(n+k) - a(n) ) is divisible by k; in this case the sequence 2*a(n) taken modulo k would be periodic with period dividing k. (End)

a(n) is the number of colored permutations by number of divisors, that is, permutations whose decomposition into a product of cycles gives the result that each cycle carries a label that is a divisor of its corresponding length. - Ricardo Gómez Aíza, Mar 08 2023

Examples

			For n = 3, there are 6 permutations that written as product of cycles are (1)(2)(3), (1)(23), (2)(13), (3)(12), (123), (132). Cycles of length one can only carry the label 1. Cycles of length two can carry the label either 1 or 2. Cycles of length three can carry the label either 1 or 3. Then a(3) = 11. - _Ricardo Gómez Aíza_, Mar 08 2023
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..445
• Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000005, A168243.

Programs

Mathematica

nmax=20; CoefficientList[Series[Product[1/(1-x^k)^(1/k),{k,1,nmax}],{x,0,nmax}],x] * Range[0,nmax]! (* Vaclav Kotesovec, May 28 2015 *)
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 26 2019 *)

Maxima

a(n):=if n=0 then 1 else (n-1)!*sum(length(divisors(i+1))*a(n-i-1)/(n-i-1)!,i,0,n-1); /* Vladimir Kruchinin, Feb 27 2015 */

Formula

This is an expansion as an exponential generating function, i.e., as sum a(n)*x^n/n!.

Equivalently, a(n)/n! is the Euler transform of [1, 1/2, 1/3, 1/4, ...].

a(n) = (n-1)!*Sum_{i=0..n-1} d(i+1)*a(n-i-1)/(n-i-1)!, a(0)=1, where d(i) is number of divisors function. - Vladimir Kruchinin, Feb 27 2015

Conjecture: log(a(n)/n!) ~ log(2)/2 * log(n)^2. - Vaclav Kotesovec, Sep 15 2018

From Ricardo Gómez Aíza, Mar 08 2023: (Start)

The above conjecture is incorrect:

a(n)/n! ~ (w(n) / n)^(1 - gamma)/sqrt(2 * Pi * abs(log(w(n) / n))) * exp(c + w(n) + (log(w(n) / n))^2 / 2), where w(n) = W(e^gamma * n), W is the Lambert W function, gamma is the Euler-Mascheroni constant, c = Pi^2 / 12 - gamma^2 / 2 - 2 * gamma(1), and gamma(1) is the 1st Stieltjes number.

log(a(n)/n!) ~ (1/2) * log(n)^2. (End)

Extensions

Edited by Franklin T. Adams-Watters, Jul 03 2009