A244495 Number of 3 X 3 matrices of nonnegative integer entries with all row and column sums <= n.

1, 34, 451, 3380, 17531, 70466, 235014, 679722, 1757085, 4147792, 9084361, 18683314, 36421463, 67798940, 121239308, 209285436, 350158809, 569759574, 904194895, 1402934104, 2132700691, 3182223374, 4667981330, 6741092150, 9595505205, 13477677876, 18697927509, 25643668006, 34794756655

Offset: 0

Author

N. J. A. Sloane, Jul 06 2014

Keywords

• nonn

Examples

			a(1)=34:
0 1's: 1,
1 1: 9,
2 1's: 3*3*2 = 18,
3 1's: 6 (transversals),
total = 34.
		

References

• Stanley, Richard P., Linear homogeneous Diophantine equations and magic labelings of graphs. Duke Math. J. 40 (1973), 607-632.
• Stanley, Richard P., Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings. Duke Math. J. 43 (1976), no. 3, 511-531.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]

Programs

Magma

[1+(25/6)*k+(3337/420)*k^2+(13777/1512)*k^3+(3289/480)*k^4+(9983/2880)*k^5+(281/240)*k^6+(73/288)*k^7+(107/3360)*k^8+(107/60480)*k^9 : k in [0..30]]; // Wesley Ivan Hurt, Jul 06 2014

Maple

f:= k -> 1+(25/6)*k+(3337/420)*k^2+(13777/1512)*k^3+(3289/480)*k^4+(9983/2880)*k^5+(281/240)*k^6+(73/288)*k^7+(107/3360)*k^8+(107/60480)*k^9:
seq(f(k),k=0..1000); # Robert Israel, Jul 06 2014

Mathematica

CoefficientList[Series[(1 + 24*x + 156*x^2 + 280*x^3 + 156*x^4 + 24*x^5 + x^6)/(1 - x)^10, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 06 2014 *)

Formula

G.f.: (1+24*x+156*x^2+280*x^3+156*x^4+24*x^5+x^6)/(1-x)^10.

a(k) = 1+(25/6)*k+(3337/420)*k^2+(13777/1512)*k^3+(3289/480)*k^4+(9983/2880)*k^5+(281/240)*k^6+(73/288)*k^7+(107/3360)*k^8+(107/60480)*k^9. - Robert Israel, Jul 06 2014