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%I M3084 N1250 #119 Oct 27 2023 18:25:44
%S 1,1,3,21,282,6210,202410,9135630,545007960,41514583320,3930730108200,
%T 452785322266200,62347376347779600,10112899541133589200,
%U 1908371363842760216400,414517594539154672566000,102681435747106627787376000,28772944645196614863048048000
%N Number of n X n matrices with nonnegative entries and every row and column sum 2.
%C Or, number of labeled 2-regular pseudodigraphs (multiple arcs and loops allowed) of order n.
%C Also, number of permutations of the multiset {1^2,2^2,...,n^2} with the descent set consisting of multiples of 2. - _Max Alekseyev_, Apr 28 2014
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, #25, a_n.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, section 3.5.10.
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (a).
%D 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 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.
%H Alois P. Heinz, <a href="/A000681/b000681.txt">Table of n, a(n) for n = 0..250</a> (first 49 terms from R. W. Robinson)
%H H. Anand, V. C. Dumir and H. Gupta, <a href="http://dx.doi.org/10.1215/S0012-7094-66-03391-6">A combinatorial distribution problem</a>, Duke Math. J., 33 (1996), 757-769.
%H Esther M. Banaian, <a href="http://digitalcommons.csbsju.edu/honors_thesis/24">Generalized Eulerian Numbers and Multiplex Juggling Sequences</a>, (2016). All College Thesis Program. Paper 24.
%H E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce <a href="http://arxiv.org/abs/1508.03673">A generalization of Eulerian numbers via rook placements</a>, arXiv:1508.03673 [math.CO], 2015.
%H S. Cockburn and J. Lesperance, <a href="http://www.jstor.org/stable/10.4169/math.mag.86.2.097">Deranged socks</a>, Mathematics Magazine, 86 (2013), 97-109.
%H Ira Gessel, <a href="http://www.mat.univie.ac.at/~slc/opapers/s17gessel.html">Enumerative applications of symmetric functions</a>, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.
%H William George Griffiths, <a href="http://dx.doi.org/10.1007/s00026-008-0336-3">On Integer Solutions to Linear Equations</a>, Annals of Combinatorics 12:1 (2008), pp. 53-70.
%H Rui-Li Liu, Feng-Zhen Zhao, <a href="https://www.emis.de/journals/JIS/VOL21/Liu/liu19.html">New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
%H Richard J. Mathar, <a href="https://arxiv.org/abs/1903.12477">2-regular Digraphs of the Lovelock Lagrangian</a>, arXiv:1903.12477 [math.GM], 2019.
%H M. L. Stein and P. R. Stein, <a href="/A001496/a001496.pdf">Enumeration of Stochastic Matrices with Integer Elements</a>, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
%H <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>
%F Sum_{n >= 0} a(n) x^n / n!^2 = exp(x/2) / sqrt(1-x).
%F D-finite with recurrence a(n) = n^2*a(n-1) - (1/2)*n*(n-1)^2*a(n-2).
%F a(n) is asymptotic to c/sqrt(n)*(n!)^2 where c=0.93019... - _Benoit Cloitre_, Jun 25 2004
%F a(n) = sum(i=0..n, 2^(i-2*n) * C(n, i)^2 * (2*n-2*i)! * i! ).
%F a(n) = 2^(-n) * sum(i=0..n, ((n!)^2*(2*i)!) / ((i!)^2*((n-i)!*2^i)) ). - _Shanzhen Gao_, Nov 05 2007
%F In Cloitre's formula is c = exp(1/2)/sqrt(Pi) = 0.9301913671026328586. - _Vaclav Kotesovec_, Aug 12 2013
%F With c as used above by Cloitre and Kotesovec, a(n) is asymptotic to c/sqrt(n)*(n!)^2 * (1 + 2/(16*n) + 50/(16*n)^2 + 1100/(16*n)^3 + 32438/(16*n)^4 + 1185660/(16*n)^5 + 50498228/(16*n)^6 + 2438464600/(16*n)^7 + 131323987366/(16*n)^8 + 7782036656108/(16*n)^9 + 501905392385436/(16*n)^10 + ...). - _Jon E. Schoenfield_, Mar 03 2014
%F E.g.f.: 2/((2-x)*W(0)), where W(k) = 1 - (2*k+1)*x/(2-x-2*(k+1)*x/W(k+1)); (continued fraction). - _Sergei N. Gladkovskii_, Nov 25 2014
%e G.f. = 1 + x + 3*x^2 + 21*x^3 + 282*x^4 + 6210*x^5 + 202410*x^6 + 9135630*x^7 + ...
%p A000681 := proc(n)
%p coeftayl( exp(x/2)/sqrt(1-x),x=0,n) ;
%p %*(n!)^2 ;
%p end proc:
%p seq(A000681(n),n=0..10) ; # _R. J. Mathar_, May 01 2019
%t a[n_] := Sum[ ((2*i)!*n!^2) / (2^i*(i!^2*(n - i)!)), {i, 0, n}]/2^n; Table[ a[n], {n, 0, 17}] (* _Jean-François Alcover_, Dec 08 2011 *)
%t a[n_] := n!*HypergeometricPFQ[{1/2, -n}, {}, -2]/2^n; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Aug 08 2012 *)
%o (Sage) from sage.combinat.integer_matrices import IntegerMatrices
%o def a(n): return IntegerMatrices([2]*n, [2]*n).cardinality() # _Ralf Stephan_, Mar 02 2014
%o (PARI) Vec( serlaplace(serlaplace( exp(x/2)/sqrt(1-x) )) ) /* _Max Alekseyev_, Apr 28 2014 */
%Y Column k=2 of A257493.
%Y Row sums of A269742 and A307804.
%Y Row and column sums equal s: A000142 (s=1), A001500 (s=3), A172806 (s=4), A172862 (s=5), A172894 (s=6), A172919 (s=7), A172944 (s=8), A172958 (s=9).
%Y Cf. A001499, A005650, A123544.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _David W. Wilson_
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