A008764 Number of 3 X 3 symmetric stochastic matrices under row and column permutations.

1, 1, 2, 4, 6, 8, 12, 16, 21, 27, 34, 42, 52, 62, 74, 88, 103, 119, 138, 158, 180, 204, 230, 258, 289, 321, 356, 394, 434, 476, 522, 570, 621, 675, 732, 792, 856, 922, 992, 1066, 1143, 1223, 1308, 1396, 1488, 1584, 1684, 1788, 1897, 2009, 2126, 2248, 2374, 2504

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			There are 6 nonisomorphic symmetric 3 X 3 matrices with row and column sums 4:
[0 0 4] [0 1 3] [0 1 3] [0 2 2] [0 2 2] [1 1 2]
[0 4 0] [1 2 1] [1 3 0] [2 0 2] [2 1 1] [1 2 1]
[4 0 0] [3 1 0] [3 0 1] [2 2 0] [2 1 1] [2 1 1]
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

Crossrefs

Cf. A019298.

Programs

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

Magma

R:=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

Maple

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

Mathematica

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 *)

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

Sage

def A008764_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
A008764_list(60) # G. C. Greubel, Sep 10 2019

Formula

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).

a(n) = A008762(n) - A008762(n-1) + A008762(n-2). - R. J. Mathar, Aug 06 2013

Extensions

Better description and more terms from Vladeta Jovovic, Feb 06 2000