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.

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, 1, 1, 28, 386, 1837, 1837, 386, 28, 1, 1, 36, 706, 5776, 12951, 5776, 706, 36, 1, 1, 45, 1191, 15085, 66021, 66021, 15085, 1191, 45, 1, 1, 55, 1889, 34399, 258355, 551681, 258355, 34399, 1889, 55, 1

Offset

0,5

Comments

Row sums give A131236.

References

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.65(a).

Links

Table of n, a(n) for n=0..65.

Formula

G.f. column 2: (-1-x-6*x^2+x^3+x^4)/(x-1)^5. - R. J. Mathar, Mar 20 2018

T(n,2) = (4+8*n+5*n^2+6*n^3+n^4)/4. - R. J. Mathar, Mar 20 2018

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

T(n,3) = (8+58*n^2+3*n^3+n^4+9*n^5+n^6)/8. - R. J. Mathar, Mar 20 2018

Example

1;
1,1;
1,3,1;
1,6,6,1;
1,10,26,10,1;
1,15,79,79,15,1;
1,21,189,451,189,21,1;
...
or as a symmetric array
1   1    1   1   1  1 1 ...
1   3    6  10  15 21 ...
1   6   26  79 189 ..
1  10   79 451 ..
1  15  189 ..
1  21 ..

Maple

A131235 := proc(m, n)
   exp((x*y*(3-x*y)+(x+y)*(2-x*y))/2/(1-x*y))/sqrt(1-x*y) ;
   coeftayl(%, y=0, n)*n!;
   coeftayl(%, x=0, m)*m! ;
end proc: # R. J. Mathar, Mar 20 2018

Mathematica

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!&];
Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 14 2023, after R. J. Mathar *)

Crossrefs

Cf. A049088 (diagonal), A131236, A131237, A088699 and A086885 (sums <= 1), A000217 (column 1)

Sequence in context: A100862 A098568 A180959 * A202812 A157243 A146769

Adjacent sequences: A131232 A131233 A131234 * A131236 A131237 A131238

Keyword

nonn,tabl

Author

Vladeta Jovovic, Jun 20 2007

Status

approved