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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.

%I A028342 #62 Oct 30 2024 04:10:19
%S A028342 1,1,3,11,59,339,2629,20677,202089,2066201,24322931,296746251,
%T A028342 4193572723,59806188571,954679763829,15845349818789,285841314451409,
%U A028342 5293203821406897,106976406006818659,2201383054398314251
%N 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.
%C A028342 From _Peter Bala_, Nov 14 2017: (Start)
%C A028342 It appears that the sequence taken modulo 10 is periodic with period 10. More generally, we conjecture
%C A028342 (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.
%C A028342 (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.
%C A028342 (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)
%C A028342 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
%H A028342 Vaclav Kotesovec, <a href="/A028342/b028342.txt">Table of n, a(n) for n = 0..445</a>
%H A028342 Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://doi.org/10.1007/s44007-024-00134-w">A unified treatment of families of partition functions</a>, La Matematica (2024). Preprint available as <a href="https://arxiv.org/abs/2303.02240">arXiv:2303.02240</a> [math.CO], 2023.
%H A028342 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A028342 This is an expansion as an exponential generating function, i.e., as sum a(n)*x^n/n!.
%F A028342 Equivalently, a(n)/n! is the Euler transform of [1, 1/2, 1/3, 1/4, ...].
%F A028342 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
%F A028342 Conjecture: log(a(n)/n!) ~ log(2)/2 * log(n)^2. - _Vaclav Kotesovec_, Sep 15 2018
%F A028342 From _Ricardo Gómez Aíza_, Mar 08 2023: (Start)
%F A028342 The above conjecture is incorrect:
%F A028342 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.
%F A028342 log(a(n)/n!) ~ (1/2) * log(n)^2. (End)
%e A028342 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
%t A028342 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 *)
%t A028342 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 *)
%t A028342 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 *)
%o A028342 (Maxima)
%o A028342 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 */
%Y A028342 Cf. A000005, A168243.
%K A028342 nonn,easy
%O A028342 0,3
%A A028342 _Jeffrey Shallit_
%E A028342 Edited by _Franklin T. Adams-Watters_, Jul 03 2009