A259786 Number T(n,k) of n X n Tesler matrices of nonnegative integers with element sum n+k; triangle T(n,k), n>=1, 0<=k<=n*(n-1)/2, read by rows.
1, 1, 1, 1, 3, 2, 1, 1, 6, 11, 11, 7, 3, 1, 1, 10, 35, 65, 81, 71, 50, 27, 12, 4, 1, 1, 15, 85, 260, 526, 771, 878, 811, 627, 416, 238, 118, 50, 18, 5, 1, 1, 21, 175, 805, 2436, 5362, 9123, 12568, 14465, 14289, 12345, 9483, 6534, 4071, 2297, 1176, 542, 224, 81, 25, 6, 1
Offset: 1
Examples
Triangle T(n,k) begins: 1; 1, 1; 1, 3, 2, 1; 1, 6, 11, 11, 7, 3, 1; 1, 10, 35, 65, 81, 71, 50, 27, 12, 4, 1; 1, 15, 85, 260, 526, 771, 878, 811, 627, 416, 238, 118, 50, 18, 5, 1; ...
Links
- Alois P. Heinz, Rows n = 1..17, flattened
Programs
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Maple
b:= proc(n, i, l) option remember; (m-> `if`(m=0, 1, expand( `if`(i=0, x^(l[1]+1)*b(l[1]+1, m-1, subsop(1=NULL, l)), add( b(n-j, i-1, subsop(i=l[i]+j, l)), j=0..n)))))(nops(l)) end: T:= n->(p->seq(coeff(p, x, i), i=n-1..degree(p)))(b(1, n-1, [0$(n-1)])): seq(T(n), n=1..8); -
Mathematica
b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, 1, Expand[ If[i == 0, x^(l[[1]] + 1)*b[l[[1]] + 1, m - 1, ReplacePart[l, 1 -> Nothing]], Sum[b[n - j, i - 1, ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]]][Length[l]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n - 1, Exponent[p, x]}]][b[1, n - 1, Table[0, {n - 1}]]]; Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)
Formula
Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A259787(n).
Comments