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

0,3

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).

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

Example

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]

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 *)

Prog

(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
(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
(Sage)
def A008764_list(prec):
    P.<x> = 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
(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

Crossrefs

Cf. A019298.

Sequence in context: A117146 A061553 A138934 * A228354 A065386 A048951

Adjacent sequences: A008761 A008762 A008763 * A008765 A008766 A008767

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved