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

1,2

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

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

Example

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.

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 *)

Prog

(Maxima)
S(n, m):=(if n=m then 1 else if n<m then 0 else S(n, m+1)+S(n-m, m));
P(n, m):=(if n=m then (n-1)! else if n<m then 0 else P(n, m+1)+S(n-m, m)*n!/(m)+(n!)/(n-m)!*P(n-m, m));
makelist(P(n, 1), n, 1, 18); /* Vladimir 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

Crossrefs

Column 1 of A210590.

Cf. A103738.

Sequence in context: A192463 A243952 A094856 * A356336 A352294 A182018

Adjacent sequences: A057620 A057621 A057622 * A057624 A057625 A057626

Keyword

nonn

Author

Leroy Quet, Oct 09 2000

Status

approved