A001501 - OEIS

This site is supported by donations to The OEIS Foundation.

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 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,5`
	REFERENCES	`L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,3).` `Shanzhen Gao, Sequences Arising from Integer Matrix Enumeration (in preparation) [From shanzhen gao (shanzhengao(AT)yahoo.com), Feb 19 2010]` `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.` `Wang, Bo-Ying; Zhang, Fuzhen. 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`
	LINKS	`T. D. Noe, Table of n, a(n) for n=0..50` `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 * (3n-3a-2b)! / (a! b! * (n-a-b)!^2 * 6^(n-a-b)). [From shanzhen gao (shanzhengao(AT)yahoo.com), Feb 19 2010]`
	MAPLE	`a:= n-> n!^2/6^n add (add ((-1)^b 2^a 3^b (3n-3a-2b)!/` `(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`
	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 *)`
	PROG	`(PARI) a(n)=local(k); if(n<0, 0, n!^2sum(j=0, n, sum(i=0, n-j, if(1, k=n-i-j; (j+3k)!/(3^i36^ki!*k!^2)))/j!/(-2)^j)) (from Michael Somos)`
	CROSSREFS	`Sequence in context: A002671 A225220 A194472 * 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`

Content is available under The OEIS End-User License Agreement .

Last modified June 20 00:11 EDT 2013. Contains 226416 sequences.