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

Author

Ralf Stephan, Mar 27 2005

Keywords

• nonn

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

Crossrefs

Row sums of A323128.

Cf. A000110, A000670.

Programs

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 *)

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

Extensions

Corrected by Vladeta Jovovic, Sep 08 2006

Offset corrected by Vaclav Kotesovec, May 03 2015