

A058528


Number of n X n (0,1) matrices with all column and row sums equal to 4.


6



1, 0, 0, 0, 1, 120, 67950, 68938800, 116963796250, 315031400802720, 1289144584143523800, 7722015017013984456000, 65599839591251908982712750, 769237071909157579108571190000, 12163525741347497524178307740904300
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OFFSET

0,6


COMMENTS

Further terms generated by a Mathematica program written by Gordon G. Cash, who thanks B. R. PerezSalvador, Universidad Autonoma Metropolitana Unidad Iztapalapa, Mexico, for providing the algorithm on which this program was based.
Also number of ways to arrange 4n rooks on an n X n chessboard, with no more than 4 rooks in each row and column.  Vaclav Kotesovec, Aug 04 2013
Generally (Canfield + McKay, 2004), a(n) ~ exp(1/2) * binomial(n,s)^(2*n) / binomial(n^2,s*n), or a(n) ~ sqrt(2*Pi) * exp(n*s1/2*(s1)^2) * (n*s)^(n*s+1/2) * (s!)^(2*n).  Vaclav Kotesovec, Aug 04 2013


REFERENCES

B. R. PerezSalvador, S. de los Cobos Silva, M. A. GutierrezAndrade and A. TorresChazaro, A Reduced Formula for Precise Numbers of (0,1) Matrices in a(R,S), Disc. Math., 2002, 256, 361372.


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..150, [Computed with Maple program by Doron Zeilberger, see link below. This replaces an earlier bfile computed by Vladeta Jovovic (and corrected terms 2631).]


FORMULA

a(n) = 24^{n} sum_{alpha +beta + gamma + mu + u =n}frac{3^{ gamma }(6)^{beta +u }8^{ mu }(n!)^{2}(4alpha +2 gamma + mu )!(beta +2 gamma )!}{alpha!beta! gamma! mu!u!} sum_{i=0}^{ floor (beta +2 gamma )/2 }frac{1}{24^{alpha  gamma +i}2^{beta +2 gamma i}i!(beta +2 gamma 2i)!(alpha  gamma +i)!}  Shanzhen Gao, Nov 07 2007
a(n) ~ exp(1/2)*C(n,4)^(2*n)/C(n^2,4*n), (Canfield + McKay, 2004).
a(n) ~ sqrt(Pi)*2^(2*n+3/2)*9^(n)*exp(4*n9/2)*n^(4*n+1/2).
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EXAMPLE

a(4) = 1 because there is only one possible 4 X 4 (0,1) matrix with all row and column sums equal to 4, the matrix of all 1's. a(5) = 120 = 5! because there are 5X4X3X2X1 ways of placing a zero in each successive column (row) so that it is not in the same row (column) as any previously placed.


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS

More terms from Gordon G. Cash (cash.gordon(AT)epa.gov), Oct 22 2002


STATUS

approved



