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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
OFFSET
1,2
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
KEYWORD
nonn
AUTHOR
Leroy Quet, Oct 09 2000
STATUS
approved