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A101370
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Number of zero-one matrices with n ones and no zero rows or columns.
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27
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1, 4, 24, 196, 2016, 24976, 361792, 5997872, 111969552, 2324081728, 53089540992, 1323476327488, 35752797376128, 1040367629940352, 32441861122796672, 1079239231677587264, 38151510015777089280, 1428149538870997774080, 56435732691153773665280
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OFFSET
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1,2
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COMMENTS
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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]
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REFERENCES
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Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, p. 435 (IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13), Springer, Berlin. [Rainer Rosenthal, Apr 10 2007]
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..400
P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.
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FORMULA
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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.
G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^m. - Vladeta Jovovic, Mar 25 2006
Inverse binomial transform of A007322. - Vladeta Jovovic, Aug 17 2006
G.f.: Sum_{n>=0} 1/(2-(1+x)^n)/2^(n+1). - Vladeta Jovovic, Sep 23 2006
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!. - Paul D. Hanna, Nov 07 2009
a(n) ~ n! / (2^(2+log(2)/2) * (log(2))^(2*(n+1))). - Vaclav Kotesovec, Dec 31 2013
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EXAMPLE
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a(2)=4:
[1 1] [1] [1 0] [0 1]
..... [1] [0 1] [1 0]
From Gus Wiseman, Nov 14 2018: (Start)
The a(3) = 24 matrices:
[111]
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[11][11][110][101][10][100][011][01][010][001]
[10][01][001][010][11][011][100][11][101][110]
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[1][10][10][10][100][100][01][01][010][01][010][001][001]
[1][10][01][01][010][001][10][10][100][01][001][100][010]
[1][01][10][01][001][010][10][01][001][10][100][010][100]
(End)
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MATHEMATICA
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m = 17; a670[n_] = Sum[ StirlingS2[n, k]*k!, {k, 0, n}]; Rest[ CoefficientList[ Series[ Sum[ a670[n]^2*(Log[1 + x]^n/n!), {n, 0, m}], {x, 0, m}], x]] (* Jean-François Alcover, Sep 02 2011, after g.f. *)
Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#]]&]], {n, 5}] (* Gus Wiseman, Nov 14 2018 *)
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PROG
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(GAP) P:=function(n) return Sum([1..n], x->Stirling2(n, x)*Factorial(x)); end;
(GAP) F:=function(n) return Sum([1..n], x->(-1)^(n-x)*Stirling1(n, x)*P(x)^2)/Factorial(n); end;
(PARI) {A000670(n)=sum(k=0, n, stirling(n, k, 2)*k!)}
{a(n)=polcoeff(sum(m=0, n, A000670(m)^2*log(1+x+x*O(x^n))^m/m!), n)}
/* Paul D. Hanna, Nov 07 2009 */
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CROSSREFS
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Cf. A000670 (the sequence (P(n)), A049311 (row and column permutations allowed), A120733, A122725, A135589, A283877, A321446, A321587.
Sequence in context: A305988 A219530 A291819 * A201338 A099021 A220690
Adjacent sequences: A101367 A101368 A101369 * A101371 A101372 A101373
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KEYWORD
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easy,nonn
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AUTHOR
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Peter J. Cameron, Jan 14 2005
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STATUS
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approved
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