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 A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0. (Formerly M4286 N1792) 22
 1, 0, 1, 6, 90, 2040, 67950, 3110940, 187530840, 14398171200, 1371785398200, 158815387962000, 21959547410077200, 3574340599104475200, 676508133623135814000, 147320988741542099484000, 36574751938491748341360000, 10268902998771351157327104000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Or, number of labeled 2-regular relations of order n. Also number of ways to arrange 2n rooks on an n X n chessboard, with no more than 2 rooks in each row and column (no 3 in a line). - Vaclav Kotesovec, Aug 03 2013 REFERENCES R. Bricard, L'Intermédiaire des Mathématiciens, 8 (1901), 312-313. L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,2). L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. Knuth. Reprinted in Problems in Applied Mathematics, ed. M. Klamkin, SIAM, 1990, p. 350. Gao, Shanzhen, and Matheis, Kenneth, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53. J. T. Lewis, Maximal L-free subsets of a squarefree array, Congressus Numerantium, 141 (1999), 151-155. R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976. 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, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (b). 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. J. H. van Lint and R. M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992), pp. 152-153. [The second edition is said to be a better reference.] LINKS R. W. Robinson and Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..48 from R. W. Robinson) H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769. P. Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806. L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. Sally Cockburn and Joshua Lesperance, Deranged Socks, Mathematics Magazine, Vol. 86, no. 2, April 2013, pp. 97-109. L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review, Vol. 21, No. 1 (Jan., 1979), p. 140. Solution by D. E. Knuth, SIAM Review, Vol. 22, No. 1 (Jan., 1980), pp. 101-102. M. E. Kuczma, 0-1-Matrices with Line-Sums Equal to 2, Am. Math. Month. 99 (1992) 959-961, E3419. Rui-Li Liu, Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7. 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 FORMULA 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. D-finite with recurrence 2*a(n) -2*n*(n-1)*a(n-1) -n*(n-1)^2*a(n-2)=0. a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth] a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand et al.) 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) 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. lim(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar, Sep 21 2007 a(n) = 4^(-n) * n!^2 * sum(i=0..n, (-2)^i * (2*n-2*i)! / (i!*(n-i)!^2)) ). - Shanzhen Gao, Feb 15 2010 MATHEMATICA a[n_] := (n-1)*n!*Gamma[n-1/2]*Hypergeometric1F1[2-n, 3/2-n, -1/2]/Sqrt[Pi]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 06 2011, after first formula *) PROG (PARI) a(n)=if(n<2, n==0, (n^2-n)*(a(n-1)+(n-1)/2*a(n-2))) (PARI) seq(n)={Vec(serlaplace(serlaplace(exp(-x/2 + O(x*x^n))/sqrt(1-x + O(x*x^n)))))}; \\ Andrew Howroyd, Sep 09 2018 (Haskell) a001499 n = a001499_list !! n a001499_list = 1 : 0 : 1 : zipWith (*) (drop 2 a002411_list)    (zipWith (+) (zipWith (*) [3, 5 ..] \$ tail a001499_list)                 (zipWith (*) (tail a000290_list) a001499_list)) -- Reinhard Zumkeller, Jun 02 2013 CROSSREFS Cf. A000681, A053871, A123544 (connected relations), A000986 (symmetric matrices), A007107 (traceless matrices). Cf. A000290, A002411, A005408. Cf. A001501. Column 2 of A008300. Row sums of A284989. Sequence in context: A002896 A266734 A004996 * A147630 A221097 A177584 Adjacent sequences:  A001496 A001497 A001498 * A001500 A001501 A001502 KEYWORD nonn,nice,easy AUTHOR STATUS approved

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Last modified July 8 22:01 EDT 2020. Contains 335537 sequences. (Running on oeis4.)