A001501 Number of n X n 0-1 matrices with all column and row sums equal to 3. (Formerly M5175 N2247)

1, 0, 0, 1, 24, 2040, 297200, 68938800, 24046189440, 12025780892160, 8302816499443200, 7673688777463632000, 9254768770160124288000, 14255616537578735986867200, 27537152449960680597739468800, 65662040698002721810659005184000

Offset

0,5

Comments

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

References

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..185 (first 51 terms from T. D. Noe)

R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)

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. [Annotated scanned copy]

Bo-Ying Wang, Fuzhen Zhang, On the precise number of (0,1)-matrices in A(R,S), Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From N. J. A. Sloane, Jun 07 2012

Index entries for sequences related to binary matrices

Formula

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

Example

G.f. = 1 + x^3 + 24*x^4 + 2040*x^5 + 297200*x^6 + 68938800*x^7 + ...

Maple

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):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 20 2011
# 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:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 13 2017

Mathematica

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;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 07 2018 *)

Prog

(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 */

Crossrefs

Cf. A001499. Column 3 of A008300. Row sums of A284990.

Sequence in context: A327197 A194472 A246602 * A054005 A107675 A173115

Adjacent sequences: A001498 A001499 A001500 * A001502 A001503 A001504

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Additional comments from Michael Somos, May 28 2002

Status

approved