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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. 34
1, 1, 3, 11, 59, 339, 2629, 20677, 202089, 2066201, 24322931, 296746251, 4193572723, 59806188571, 954679763829, 15845349818789, 285841314451409, 5293203821406897, 106976406006818659, 2201383054398314251 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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)

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..445

N. J. A. Sloane, Transforms

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

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 *)

PROG

(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 */

CROSSREFS

Cf. A000005.

Sequence in context: A319248 A290484 A156560 * A137982 A193364 A074509

Adjacent sequences:  A028339 A028340 A028341 * A028343 A028344 A028345

KEYWORD

nonn,easy

AUTHOR

Jeffrey Shallit

EXTENSIONS

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

STATUS

approved

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Last modified January 28 10:02 EST 2020. Contains 331319 sequences. (Running on oeis4.)