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A001501
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Number of n X n 0-1 matrices with all column and row sums equal to 3.
(Formerly M5175 N2247)
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15
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1, 0, 0, 1, 24, 2040, 297200, 68938800, 24046189440, 12025780892160, 8302816499443200, 7673688777463632000, 9254768770160124288000, 14255616537578735986867200, 27537152449960680597739468800, 65662040698002721810659005184000
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OFFSET
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0,5
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COMMENTS
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Also, for n >= 3, number of bicubical graphs on 2n labeled nodes of two colors [Read, 1958, 1971] - N. J. A. Sloane, Sep 08 2014
Also number of ways to arrange 3n rooks on an n X n chessboard, with no more than 3 rooks in each row and column (no 4 in a line). - Vaclav Kotesovec, Aug 03 2013
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,3).
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Example 1.1.3, page 2, f(n).
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.
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LINKS
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FORMULA
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a(n) = n!^2/6^n * Sum_{a=0..n} Sum_{b=0..n-a} (-1)^b * 2^a * 3^b * (3*n-3*a-2*b)! / (a! * b! * (n-a-b)!^2 * 6^(n-a-b)). - Shanzhen Gao, Feb 19 2010
D-finite with recurrence: 12*(3*n-5)*a(n) = 9*n*(3*n^2-5*n+4)*(n-1)*a(n-1) + 3*(n-2)*n*(3*n+1)*(n-1)^2*a(n-2) + (n-2)^2*n*(9*n^2-30*n+13)*(n-1)^2*a(n-3) - (n-3)^2*(n-2)^2*n*(3*n-2)*(n-1)^2*a(n-4). - Vaclav Kotesovec, Aug 03 2013
a(n) ~ sqrt(6*Pi) * (3/4)^n * n^(3*n+1/2) / exp(3*n+2). - Vaclav Kotesovec, Aug 03 2013
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EXAMPLE
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G.f. = 1 + x^3 + 24*x^4 + 2040*x^5 + 297200*x^6 + 68938800*x^7 + ...
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MAPLE
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a:= n-> n!^2/6^n *add(add((-1)^b *2^a *3^b *(3*n-3*a-2*b)!/
(a! *b! *(n-a-b)!^2 *6^(n-a-b)), b=0..n-a), a=0..n):
# second Maple program:
a:= proc(n) option remember; `if`(n<4, (n-1)*(n-2)/2,
n*(n-1)*(9*(3*n^2-5*n+4)*a(n-1)+(3*n-6)*(3*n+1)*
(n-1)*a(n-2)+(9*n^2-30*n+13)*(n-1)*(n-2)^2*a(n-3)
-(3*n-2)*(n-1)*(n-2)^2*(n-3)^2*a(n-4))/(36*n-60))
end:
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MATHEMATICA
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Table[6^(-n) Total[Map[(-1)^#[[2]] n!^2 (#[[2]] + 3 #[[3]])! 2^#[[1]] 3^#[[2]]/(#[[1]]! #[[2]]! #[[3]]!^2 6^#[[3]]) &, Compositions[n, 3]]], {n, 0, 20}] (* Geoffrey Critzer, Mar 19 2011 *)
a[n_] := n!^2*Sum[2^(2k-n)*3^(k-n)*(3(n-k))!*HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k )!^2), {k, 0, n}]/6^n;
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PROG
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(PARI) {a(n) = local(k); if( n<0, 0, n!^2 * sum(j=0, n, sum(i=0, n-j, if(1, k=n-i-j; (j + 3*k)! / (3^i * 36^k * i! * k!^2))) / (j! * (-2)^j)))}; /* Michael Somos, May 28 2002 */
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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