|
|
A008764
|
|
Number of 3 X 3 symmetric stochastic matrices under row and column permutations.
|
|
1
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
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).
|
|
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)
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1+x^3)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()
(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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|