Search: id:A101370
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%I A101370
%S A101370 1,4,24,196,2016,24976,361792,5997872,111969552,2324081728,53089540992,
%T A101370 1323476327488,35752797376128,1040367629940352,32441861122796672,
%U A101370 1079239231677587264,38151510015777089280
%N A101370 Number of zero-one matrices with n ones and no zero rows or columns.
%C A101370 a(n) = (1/(4n!)) * Sum_{r, s>=0} (rs)_n / 2^(r+s) }, where (m)_n is the 
               falling factorial m * (m-1) * ... * (m-n+1). [Maia and Mendez]
%D A101370 Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen 
               Inhalts, p. 435 (IV, 4. Mitteilungen zur Lehre vom Transfiniten, 
               VIII Nr. 13), Springer, Berlin.
%H A101370 P. J. Cameron, D. A. Gewurz and F. Merola, <a href="http://www.maths.qmw.ac.uk/
               ~pjc/preprints/product.pdf">Product action</a>, Discrete Math., 308 
               (2008), 386-394.
%H A101370 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">
               On the arithmetic product of combinatorial species</a>
%F A101370 a(n) = (Sum s(n, k) * P(k)^2)/n!, where P(n) is the number of labeled 
               total preorders on {1, ..., n} (A000670), s are signed Stirling numbers 
               of the first kind.
%F A101370 G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^m. 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 25 2006
%F A101370 Inverse binomial transform of A007322. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Aug 17 2006
%F A101370 G.f.: Sum_{n>=0} 1/(2-(1+x)^n)/2^(n+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Sep 23 2006
%F A101370 G.f.: Sum_{n>=0} A000670(n)^2*log(1+x)^n/n! where 1/(1-x) = Sum_{n>=0} 
               A000670(n)*log(1+x)^n/n!. [From Paul D. Hanna (pauldhanna(AT)juno.com), 
               Nov 07 2009]
%e A101370 a(2)=4:
%e A101370 [1 1] [1] [1 0] [0 1]
%e A101370 ..... [1] [0 1] [1 0]
%o A101370 (GAP) P:=function(n) return Sum([1..n],x->Stirling2(n,x)*Factorial(x)); 
               end;
%o A101370 (GAP) F:=function(n) return Sum([1..n],x->(-1)^(n-x)*Stirling1(n,x)*P(x)^2)/
               Factorial(n); end;
%o A101370 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 07 2009: 
               (Start)
%o A101370 (PARI) {Stirling2(n, k)=if(k<0|k>n,0, sum(i=0,k,(-1)^i*binomial(k, i)/
               k!*(k-i)^n))}
%o A101370 {A000670(n)=sum(k=0,n,Stirling2(n, k)*k!)}
%o A101370 {a(n)=polcoeff(sum(m=0,n,A000670(m)^2*log(1+x+x*O(x^n))^m/m!),n)} (End)
%Y A101370 Cf. A000670 (the sequence (P(n)).
%Y A101370 Cf. A049311 (row and column permutations allowed).
%Y A101370 Cf. A000670, A122725. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 
               07 2009]
%Y A101370 Sequence in context: A073840 A024249 A007145 this_sequence A099021 A136229 
               A138419
%Y A101370 Adjacent sequences: A101367 A101368 A101369 this_sequence A101371 A101372 
               A101373
%K A101370 easy,nonn
%O A101370 1,2
%A A101370 Peter J. Cameron (p.j.cameron(AT)qmul.ac.uk), Jan 14 2005
%E A101370 Cantor reference from Rainer Rosenthal, Apr 10 2007

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