%I M5175 N2247 #78 Oct 27 2023 19:26:21
%S 1,0,0,1,24,2040,297200,68938800,24046189440,12025780892160,
%T 8302816499443200,7673688777463632000,9254768770160124288000,
%U 14255616537578735986867200,27537152449960680597739468800,65662040698002721810659005184000
%N Number of n X n 0-1 matrices with all column and row sums equal to 3.
%C 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
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,3).
%D R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Example 1.1.3, page 2, f(n).
%D 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.
%H Alois P. Heinz, <a href="/A001501/b001501.txt">Table of n, a(n) for n = 0..185</a> (first 51 terms from T. D. Noe)
%H R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a> (gives initial terms of this sequence)
%H M. L. Stein and P. R. Stein, <a href="/A001496/a001496.pdf">Enumeration of Stochastic Matrices with Integer Elements</a>, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
%H Bo-Ying Wang, Fuzhen Zhang, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00197-0">On the precise number of (0,1)-matrices in A(R,S)</a>, Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From _N. J. A. Sloane_, Jun 07 2012
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F 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
%F 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
%F a(n) ~ sqrt(6*Pi) * (3/4)^n * n^(3*n+1/2) / exp(3*n+2). - _Vaclav Kotesovec_, Aug 03 2013
%e G.f. = 1 + x^3 + 24*x^4 + 2040*x^5 + 297200*x^6 + 68938800*x^7 + ...
%p a:= n-> n!^2/6^n *add(add((-1)^b *2^a *3^b *(3*n-3*a-2*b)!/
%p (a! *b! *(n-a-b)!^2 *6^(n-a-b)), b=0..n-a), a=0..n):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 20 2011
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n<4, (n-1)*(n-2)/2,
%p n*(n-1)*(9*(3*n^2-5*n+4)*a(n-1)+(3*n-6)*(3*n+1)*
%p (n-1)*a(n-2)+(9*n^2-30*n+13)*(n-1)*(n-2)^2*a(n-3)
%p -(3*n-2)*(n-1)*(n-2)^2*(n-3)^2*a(n-4))/(36*n-60))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 13 2017
%t 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 *)
%t 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;
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 07 2018 *)
%o (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 */
%Y Cf. A001499. Column 3 of A008300. Row sums of A284990.
%K nonn,nice
%O 0,5
%A _N. J. A. Sloane_
%E Additional comments from _Michael Somos_, May 28 2002