cp's OEIS Frontend

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

%I A104600 #19 Sep 03 2019 11:23:03
%S A104600 1,1,5,49,795,18881,611193,25704253,1356235163,87419692453,
%T A104600 6741175388313,611464105166993,64336296019640307,7760748741918246361,
%U A104600 1062626712168331953737,163738827988386433177093,28181351778805732986601035,5382075236937341624838444077
%N A104600 Number of matrices of any size up to column permutations, with n different elements, zero elsewhere and with no zero row or column.
%H A104600 Alois P. Heinz, <a href="/A104600/b104600.txt">Table of n, a(n) for n = 0..274</a>
%H A104600 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">On the arithmetic product of combinatorial species</a>
%F A104600 (1/(2e)) * Sum{r, s>=0, (rs)_n / [2^r s! ] }, where (m)_n is the falling factorial m * (m-1) * ... * (m-n+1).
%F A104600 E.g.f.: exp(-1)*sum(exp((1+x)^n)/2^(n+1),n=0..infinity). - _Vladeta Jovovic_, Sep 24 2006
%F A104600 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000670(k)*A000110(k). - _Vladeta Jovovic_, Sep 27 2006
%F A104600 exp(-1)*sum(1/(2-(1+x)^n)/n!,n=0..infinity) is also e.g.f. - _Vladeta Jovovic_, Oct 09 2006
%p A104600 b:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
%p A104600       *binomial(n-1, j-1)*b(n-j, k), j=1..min(k, n)))
%p A104600     end:
%p A104600 a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p A104600 seq(a(n), n=0..21);  # _Alois P. Heinz_, Sep 03 2019
%t A104600 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 *)
%t A104600 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 *)
%Y A104600 Row sums of A323128.
%Y A104600 Cf. A000110, A000670.
%K A104600 nonn
%O A104600 0,3
%A A104600 _Ralf Stephan_, Mar 27 2005
%E A104600 Corrected by _Vladeta Jovovic_, Sep 08 2006
%E A104600 Offset corrected by _Vaclav Kotesovec_, May 03 2015