A248028 a(n) = Sum_{k=0..n} |Stirling1(n, k)|*(n-k)! for n>=0.

1, 1, 2, 8, 65, 957, 22512, 773838, 36561289, 2271696241, 179538792358, 17584290721868, 2090031277816649, 296326507395472205, 49400463740287289892, 9566059122999739401954, 2129221864475839211318769, 539805407803681202368358785, 154636541536285163968515043306, 49702496963149041682740769491568

Offset

0,3

Comments

Compare to A007840(n) = Sum_{k=0..n} |Stirling1(n, k)|*k!, which equals the number of factorizations of permutations of n letters into ordered cycles.

For n > 1, a(n) is equal to the permanent of the (n-1) X (n-1) matrix in which the (i, j)-entry is equal to delta(i, j) + i, letting delta denote the Kronecker delta function, as illustrated in the below example. - John M. Campbell, Jan 21 2018

Links

Table of n, a(n) for n=0..19.

Formula

a(n) ~ ((n-1)!)^2. - Vaclav Kotesovec, Sep 30 2014

Example

For example, the (5-1) X (5-1) matrix of the form indicated above is equal to
[2 1 1 1]
[2 3 2 2]
[3 3 4 3]
[4 4 4 5]
and the permanent of the above matrix is equal to 957 = a(5). - John M. Campbell, Jan 21 2018

Mathematica

Table[Sum[Abs[StirlingS1[n, k]]*(n-k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 30 2014 *)

Prog

(PARI) {Stirling1(n, k)=if(n==0, 1, n!*polcoeff(binomial(x, n), k))}
{a(n)=sum(k=0, n, (-1)^(n-k)*Stirling1(n, k)*(n-k)!)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A008275 (Stirling1 numbers).

Sequence in context: A091794 A013134 A009602 * A011836 A309251 A100623

Adjacent sequences: A248025 A248026 A248027 * A248029 A248030 A248031

Keyword

nonn

Author

Paul D. Hanna, Sep 29 2014

Status

approved