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%I #22 Sep 08 2022 08:44:36
%S 1,1,2,4,6,8,12,16,21,27,34,42,52,62,74,88,103,119,138,158,180,204,
%T 230,258,289,321,356,394,434,476,522,570,621,675,732,792,856,922,992,
%U 1066,1143,1223,1308,1396,1488,1584,1684,1788,1897,2009,2126,2248,2374,2504
%N Number of 3 X 3 symmetric stochastic matrices under row and column permutations.
%H G. C. Greubel, <a href="/A008764/b008764.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1,-1,1,-1,2,-1).
%F Expansion of (1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = (1-x+x^2)/( (1+x)*(1+x+x^2)*(1+x^2)*(1-x)^4).
%F a(n) = A008762(n) - A008762(n-1) + A008762(n-2). - _R. J. Mathar_, Aug 06 2013
%e There are 6 nonisomorphic symmetric 3 X 3 matrices with row and column sums 4:
%e [0 0 4] [0 1 3] [0 1 3] [0 2 2] [0 2 2] [1 1 2]
%e [0 4 0] [1 2 1] [1 3 0] [2 0 2] [2 1 1] [1 2 1]
%e [4 0 0] [3 1 0] [3 0 1] [2 2 0] [2 1 1] [2 1 1]
%p seq(coeff(series((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # _G. C. Greubel_, Sep 10 2019
%t LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,4,6,8,12,16,21}, 60] (* _G. C. Greubel_, Sep 10 2019 *)
%o (PARI) my(x='x+O('x^60)); Vec((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))) \\ _G. C. Greubel_, Sep 10 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // _G. C. Greubel_, Sep 10 2019
%o (Sage)
%o def A008764_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
%o A008764_list(60) # _G. C. Greubel_, Sep 10 2019
%o (GAP) a:=[1,1,2,4,6,8,12,16,21];; for n in [10..60] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # _G. C. Greubel_, Sep 10 2019
%Y Cf. A019298.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E Better description and more terms from _Vladeta Jovovic_, Feb 06 2000
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