A000985 Number of n X n symmetric matrices with nonnegative entries and all row sums 2.
1, 1, 3, 11, 56, 348, 2578, 22054, 213798, 2313638, 27627434, 360646314, 5107177312, 77954299144, 1275489929604, 22265845018412, 412989204564572, 8109686585668956, 168051656468233972, 3664479286118269972, 83868072451846938336, 2009964340465840802576
Offset: 0
References
- 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 Example 5.2.7.
Links
- T. D. Noe, Table of n, a(n) for n=0..100
- Jacob L. Bourjaily, Michael Plesser, and Cristian Vergu, The Many Colours of Amplitudes, arXiv:2412.21189 [hep-th], 2024. See p. 33.
- P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 584
- H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), vol 3, 653-659.
- H. Gupta, Enumeration of symmetric matrices (annotated scanned copy)
- Tomislav Došlic and Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). - From _N. J. A. Sloane_, May 01 2012
Crossrefs
Cf. A000986.
Programs
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Mathematica
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 *)
Formula
E.g.f.: (1-x)^(-1/2)*exp(x^2/4 + x/(2*(1-x))).
a(n) ~ n^n*exp(sqrt(2*n)-n)/sqrt(2) * (1-5/(24*sqrt(2*n))). - Vaclav Kotesovec, Jul 29 2013
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