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

%I A244495 #18 Sep 08 2022 08:46:08
%S A244495 1,34,451,3380,17531,70466,235014,679722,1757085,4147792,9084361,
%T A244495 18683314,36421463,67798940,121239308,209285436,350158809,569759574,
%U A244495 904194895,1402934104,2132700691,3182223374,4667981330,6741092150,9595505205,13477677876,18697927509,25643668006,34794756655
%N A244495 Number of 3 X 3 matrices of nonnegative integer entries with all row and column sums <= n.
%D A244495 Stanley, Richard P., Linear homogeneous Diophantine equations and magic labelings of graphs. Duke Math. J. 40 (1973), 607-632.
%D A244495 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.
%H A244495 Robert Israel, <a href="/A244495/b244495.txt">Table of n, a(n) for n = 0..10000</a>
%H A244495 R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission]
%F A244495 G.f.: (1+24*x+156*x^2+280*x^3+156*x^4+24*x^5+x^6)/(1-x)^10.
%F A244495 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
%e A244495 a(1)=34:
%e A244495 0 1's: 1,
%e A244495 1 1: 9,
%e A244495 2 1's: 3*3*2 = 18,
%e A244495 3 1's: 6 (transversals),
%e A244495 total = 34.
%p A244495 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:
%p A244495 seq(f(k),k=0..1000); # _Robert Israel_, Jul 06 2014
%t A244495 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 *)
%o A244495 (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
%K A244495 nonn
%O A244495 0,2
%A A244495 _N. J. A. Sloane_, Jul 06 2014