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A131235 Triangle read by rows: T(n,k) is number of (n-k) X k matrices, k=0..n, with nonnegative integer entries and every row and column sum <= 2.

%I A131235 #23 Sep 14 2023 08:01:06
%S A131235 1,1,1,1,3,1,1,6,6,1,1,10,26,10,1,1,15,79,79,15,1,1,21,189,451,189,21,
%T A131235 1,1,28,386,1837,1837,386,28,1,1,36,706,5776,12951,5776,706,36,1,1,45,
%U A131235 1191,15085,66021,66021,15085,1191,45,1,1,55,1889,34399,258355,551681,258355,34399,1889,55,1
%N A131235 Triangle read by rows: T(n,k) is number of (n-k) X k matrices, k=0..n, with nonnegative integer entries and every row and column sum <= 2.
%C A131235 Row sums give A131236.
%D A131235 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.65(a).
%F A131235 G.f. column 2: (-1-x-6*x^2+x^3+x^4)/(x-1)^5. -  _R. J. Mathar_, Mar 20 2018
%F A131235 T(n,2) = (4+8*n+5*n^2+6*n^3+n^4)/4. -  _R. J. Mathar_, Mar 20 2018
%F A131235 G.f. column 3: -(1+3*x+30*x^2+73*x^3+24*x^4-48*x^5+7*x^6)/(x-1)^7 . -  _R. J. Mathar_, Mar 20 2018
%F A131235 T(n,3) = (8+58*n^2+3*n^3+n^4+9*n^5+n^6)/8. - _R. J. Mathar_, Mar 20 2018
%e A131235 1;
%e A131235 1,1;
%e A131235 1,3,1;
%e A131235 1,6,6,1;
%e A131235 1,10,26,10,1;
%e A131235 1,15,79,79,15,1;
%e A131235 1,21,189,451,189,21,1;
%e A131235 ...
%e A131235 or as a symmetric array
%e A131235 1   1    1   1   1  1 1 ...
%e A131235 1   3    6  10  15 21 ...
%e A131235 1   6   26  79 189 ..
%e A131235 1  10   79 451 ..
%e A131235 1  15  189 ..
%e A131235 1  21 ..
%p A131235 A131235 := proc(m,n)
%p A131235    exp((x*y*(3-x*y)+(x+y)*(2-x*y))/2/(1-x*y))/sqrt(1-x*y) ;
%p A131235    coeftayl(%,y=0,n)*n!;
%p A131235    coeftayl(%,x=0,m)*m! ;
%p A131235 end proc: # _R. J. Mathar_, Mar 20 2018
%t A131235 T[n_, k_] := Module[{ex}, ex = Exp[(x*y*(3 - x*y) + (x + y)*(2 - x*y))/2/(1 - x*y)]/Sqrt[1 - x*y]; SeriesCoefficient[ex, {y, 0, k}]*k! // SeriesCoefficient[#, {x, 0, n}]*n!&];
%t A131235 Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 14 2023, after _R. J. Mathar_ *)
%Y A131235 Cf. A049088 (diagonal), A131236, A131237, A088699 and A086885 (sums <= 1), A000217 (column 1)
%K A131235 nonn,tabl
%O A131235 0,5
%A A131235 _Vladeta Jovovic_, Jun 20 2007