%I A000681 M3084 N1250
%S A000681 1,1,3,21,282,6210,202410,9135630,545007960,41514583320,3930730108200,
%T A000681 452785322266200,62347376347779600,10112899541133589200,1908371363842760216400,
%U A000681 414517594539154672566000,102681435747106627787376000,28772944645196614863048048000
%N A000681 Number of n X n matrices with nonnegative entries and every row and column 
               sum 2.
%C A000681 Or, number of labeled 2-regular pseudodigraphs (multiple arcs and loops 
               allowed) of order n.
%D A000681 H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, 
               Duke Math. J., 33 (1996), 757-769.
%D A000681 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, #25, a_n.
%D A000681 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley 
               and Sons, N.Y., 1983, section 3.5.10.
%D A000681 William George Griffiths, "On Integer Solutions to Linear Equations", 
               Annals of Combinatorics 12:1 (2008), pp. 53-70. [From Charles R Greathouse 
               IV Apr 03 2009]
%D A000681 R. W. Robinson, Numerical implementation of graph counting algorithms, 
               AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
%D A000681 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000681 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000681 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Cor. 5.5.11 (a).
%D A000681 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.
%D A000681 C. B. Tompkins, Methods of successive restrictions in computational problems 
               involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. 
               XV pp. 95-106; Amer. Math. Soc., Providence, R.I.
%D A000681 Zhonghua,Tan and Shanzhen Gao, Counting (0,1,2)-Matrices, submitted.
%H A000681 R. W. Robinson, <a href="b000681.txt">Table of n, a(n) for n = 0..48</
               a>
%H A000681 Ira Gessel, <a href="http://www.mat.univie.ac.at/~slc/opapers/s17gessel.html">
               Enumerative applications of symmetric functions</a>
%H A000681 <a href="Sindx_Mag.html#magic">Index entries for sequences related to 
               magic squares</a>
%F A000681 Sum_{n >= 0} a(n) x^n / n!^2 = 1/(1-x)^(1/2)*exp(1/2*x).
%F A000681 a(n) = n^2*a(n-1) - (1/2)*n*(n-1)^2*a(n-2).
%F A000681 a(n) is asymptotic to c/sqrt(n)*(n!)^2 where c=0.93019... - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jun 25 2004
%F A000681 Sum[i=0..n, 2^(i-2n) * C(n, i)^2 * (2n-2i)! * i! ].
%F A000681 a(n)=2^{-n} sum_{i=0}^{n}\frac{(n!)^{2}(2i)!}{(i!)^{2}((n-i)!2^{i})} 
               - Shanzhen Gao (sgao2(AT)fau.edu), Nov 05 2007
%Y A000681 Cf. A001499, A005650, A123544.
%Y A000681 Sequence in context: A066206 A130032 A126461 this_sequence A055555 A158888 
               A005329
%Y A000681 Adjacent sequences: A000678 A000679 A000680 this_sequence A000682 A000683 
               A000684
%K A000681 nonn,nice,easy
%O A000681 0,3
%A A000681 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A000681 More terms from David W. Wilson (davidwwilson(AT)comcast.net)