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%I M2907 N1168 #39 Mar 15 2023 11:54:40
%S 1,1,3,11,56,348,2578,22054,213798,2313638,27627434,360646314,
%T 5107177312,77954299144,1275489929604,22265845018412,412989204564572,
%U 8109686585668956,168051656468233972,3664479286118269972,83868072451846938336,2009964340465840802576
%N Number of n X n symmetric matrices with nonnegative entries and all row sums 2.
%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 Example 5.2.7.
%H T. D. Noe, <a href="/A000985/b000985.txt">Table of n, a(n) for n=0..100</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 584
%H H. Gupta, <a href="http://dx.doi.org/10.1215/S0012-7094-68-03567-9">Enumeration of symmetric matrices</a>, Duke Math. J., 35 (1968), vol 3, 653-659.
%H H. Gupta, <a href="/A000085/a000085.pdf">Enumeration of symmetric matrices</a> (annotated scanned copy)
%H Tomislav Došlic, Darko Veljan, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.066">Logarithmic behavior of some combinatorial sequences</a>, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). - From _N. J. A. Sloane_, May 01 2012
%F E.g.f.: (1-x)^(-1/2)*exp(x^2/4 + x/(2*(1-x))).
%F a(n) ~ n^n*exp(sqrt(2*n)-n)/sqrt(2) * (1-5/(24*sqrt(2*n))). - _Vaclav Kotesovec_, Jul 29 2013
%F Recurrence: 2*a(n) = 2*(2*n-1)*a(n-1) - 2*(n-2)*(n-1)*a(n-2) - 2*(n-2)*(n-1)*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Jul 29 2013
%t max = 21; egf[x_] := (1-x)^(-1/2)*Exp[x^2/4 + x/(2*(1-x))]; CoefficientList[ Series[ egf[x], {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Nov 25 2011 *)
%Y Cf. A000986.
%K nonn,nice,easy
%O 0,3
%A _N. J. A. Sloane_
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