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

1,5

Comments

For the definition of Tesler matrices see A008608.

Links

Alois P. Heinz, Rows n = 1..17, flattened

Formula

Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A259787(n).

Example

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;
  ...

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 *)

Crossrefs

Row sums give A008608.

Cf. A000217, A259787.

Sequence in context: A058280 A113185 A132069 * A254410 A073201 A118654

Adjacent sequences: A259783 A259784 A259785 * A259787 A259788 A259789

Keyword

nonn,tabf

Author

Alois P. Heinz, Jul 05 2015

Status

approved