A104600 Number of matrices of any size up to column permutations, with n different elements, zero elsewhere and with no zero row or column.

1, 1, 5, 49, 795, 18881, 611193, 25704253, 1356235163, 87419692453, 6741175388313, 611464105166993, 64336296019640307, 7760748741918246361, 1062626712168331953737, 163738827988386433177093, 28181351778805732986601035, 5382075236937341624838444077

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..274

M. Maia and M. Mendez, On the arithmetic product of combinatorial species

Formula

(1/(2e)) * Sum{r, s>=0, (rs)_n / [2^r s! ] }, where (m)_n is the falling factorial m * (m-1) * ... * (m-n+1).

E.g.f.: exp(-1)*sum(exp((1+x)^n)/2^(n+1),n=0..infinity). - Vladeta Jovovic, Sep 24 2006

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000670(k)*A000110(k). - Vladeta Jovovic, Sep 27 2006

exp(-1)*sum(1/(2-(1+x)^n)/n!,n=0..infinity) is also e.g.f. - Vladeta Jovovic, Oct 09 2006

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
      *binomial(n-1, j-1)*b(n-j, k), j=1..min(k, n)))
    end:
a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 03 2019

Mathematica

Table[Sum[StirlingS1[n, k] * Sum[StirlingS2[k, j]*j!, {j, 0, k}] * BellB[k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 03 2015 *)
Table[1/(2*E) * Sum[Sum[Product[r*s-k, {k, 0, n-1}] / (2^r s!), {r, 0, Infinity}], {s, 0, Infinity}], {n, 0, 10}] (* Vaclav Kotesovec, May 03 2015 *)

Crossrefs

Row sums of A323128.

Cf. A000110, A000670.

Sequence in context: A290755 A062995 A293847 * A221972 A002111 A305114

Adjacent sequences: A104597 A104598 A104599 * A104601 A104602 A104603

Keyword

nonn

Author

Ralf Stephan, Mar 27 2005

Extensions

Corrected by Vladeta Jovovic, Sep 08 2006

Offset corrected by Vaclav Kotesovec, May 03 2015

Status

approved