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A057623 a(n) = n! * (sum of reciprocals of all parts in unrestricted partitions of n). 4
1, 5, 29, 218, 1814, 18144, 196356, 2427312, 32304240, 475637760, 7460546400, 127525829760, 2302819079040, 44659367020800, 911770840108800, 19784985947596800, 449672462639769600, 10790180876185804800, 270071861749240320000, 7094011359005190144000 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 * A182018 A309260 A087662

Adjacent sequences:  A057620 A057621 A057622 * A057624 A057625 A057626

KEYWORD

nonn

AUTHOR

Leroy Quet, Oct 09 2000

STATUS

approved

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Last modified May 9 10:53 EDT 2021. Contains 343732 sequences. (Running on oeis4.)