A057623 a(n) = n! * (sum of reciprocals of all parts in unrestricted partitions of n).

1, 5, 29, 218, 1814, 18144, 196356, 2427312, 32304240, 475637760, 7460546400, 127525829760, 2302819079040, 44659367020800, 911770840108800, 19784985947596800, 449672462639769600, 10790180876185804800, 270071861749240320000, 7094011359005190144000

Offset: 1

Author

Leroy Quet, Oct 09 2000

Keywords

• nonn

Examples

			The unrestricted partitions of 3 are 1 + 1 + 1, 1 + 2 and 3. So a(3) = 3! *(1 + 1 + 1 + 1 + 1/2 + 1/3) = 29.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..400
• Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO]; see p.27

Crossrefs

Column 1 of A210590.

Cf. A103738.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, (p-> p+[0, p[1]/i])(b(n-i, i)))))
    end:
a:= n-> n!*b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 11 2014

Mathematica

b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[ {p}, p + {0, p[[1]]/i}][b[n-i, i]]]]]; a[n_] := n!*b[n, n][[2]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2015, after Alois P. Heinz *)
Table[n!*Sum[DivisorSigma[1, k]*PartitionsP[n - k]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, May 29 2018 *)

Maxima

S(n,m):=(if n=m then 1 else if nVladimir Kruchinin, Sep 10 2014 */

PARI

{a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-1-t)), n), 1)} \\ Seiichi Manyama, Nov 07 2020

Formula

n! *sum_{k=1 to n} [sigma(k) p(n-k) /k], where sigma(n) = sum of positive divisors of n and p(n) = number of unrestricted partitions of n.

a(n) = P(n,1), where P(n,m) = P(n,m+1)+S(n-m,m)*n!/m+n!/(n-m)!*P(n-m,m)), P(n,n)=(n-1)!, P(n,m)=0 for m>n, S(n,m) is triangle of A026807. - Vladimir Kruchinin, Sep 10 2014