Search: id:A001499
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%I A001499 M4286 N1792
%S A001499 1,0,1,6,90,2040,67950,3110940,187530840,14398171200,1371785398200,
%T A001499 158815387962000,21959547410077200,3574340599104475200,676508133623135814000,
%U A001499 147320988741542099484000,36574751938491748341360000,10268902998771351157327104000
%N A001499 Number of n X n matrices with exactly 2 1's in each row and column, other 
               entries 0.
%C A001499 Or, number of labeled 2-regular relations of order n.
%D A001499 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001499 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001499 H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, 
               Duke Math. J., 33 (1966), 757-769.
%D A001499 R. Bricard, L'Interm\'{e}diaire des Math\'{e}maticiens, 8 (1901), 312-313.
%D A001499 L. Carlitz, Enumeration of symmetric arrays, {\em Duke Math. J.}, {\bf 
               33} (1966), 771-782.
%D A001499 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,2).
%D A001499 L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. 
               Knuth. Reprinted in Problems in Applied Mathematices, ed. M. Klamkin, 
               SIAM, 1990, p. 350.
%D A001499 J. T. Lewis, Maximal $L$-free subsets of a square-free array, {\em Congressus 
               Numerantium}, {\bf 141} (1999), 151-155.
%D A001499 J. H. van Lint and R. M. Wilson, {\em A Course in Combinatorics}, (Cambridge 
               University Press, Cambridge, 1992), pp. 152-153. [The second edition 
               is said to be a better reference.]
%D A001499 R. W. Robinson, Numerical implementation of graph counting algorithms, 
               AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
%D A001499 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Cor. 5.5.11 (b).
%D A001499 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 A001499 R. W. Robinson, <a href="b001499.txt">Table of n, a(n) for n = 0..48</
               a>
%H A001499 <a href="Sindx_Mat.html#binmat">Index entries for sequences related to 
               binary matrices</a>
%H A001499 M. E. Kuczma, <a href="http://links.jstor.org/sici?sici=0002-9890%28199212%2999%3A10%3C959%3AE%3E2.0.CO%3B2-S\
               ">0-1-Matrices with Line-Sums Equal to 2</a>, Am. Math. Month. 99 
               (1992) 959-961, E3419.
%F A001499 a(n) = (n! (n-1) Gamma(n-1/2) / Gamma(1/2) ) * 1F1[2-n; 3/2-n; -1/2] 
               [Erlebach and Ruehr]. This representation is exact, asymptotic and 
               convergent.
%F A001499 a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth]
%F A001499 a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand 
               et al.)
%F A001499 Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(-x/2)/sqrt(1-x); a(n) = n(n-1)/2 [ 
               2 a(n-1) + (n-1) a(n-2) ] (Bricard)
%F A001499 b_n = a_n/n! satisfies b_n = (n-1)(b_{n-1} + b_{n-2}/2); e.g.f. for {b_n} 
               and for derangements (A000166) are related by D(x) = B(x)^2.
%F A001499 lim(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar 
               (mathar(AT)strw.leidenuniv.nl), Sep 21 2007
%o A001499 (PARI) a(n)=if(n<2,n==0,(n^2-n)*(a(n-1)+(n-1)/2*a(n-2)))
%Y A001499 Cf. A000681, A053871, A123544 (connected relations).
%Y A001499 Sequence in context: A138462 A002896 A004996 this_sequence A147630 A132467 
               A000680
%Y A001499 Adjacent sequences: A001496 A001497 A001498 this_sequence A001500 A001501 
               A001502
%K A001499 nonn,nice,easy
%O A001499 0,4
%A A001499 N. J. A. Sloane (njas(AT)research.att.com).

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