A259841 Number T(n,k) of elements k in all n X n Tesler matrices of nonnegative integers; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 3, 1, 15, 5, 2, 117, 37, 17, 7, 1367, 418, 189, 100, 40, 23329, 7027, 3058, 1688, 939, 357, 570933, 171428, 72194, 39274, 24050, 13429, 4820, 19740068, 5948380, 2449366, 1293768, 807576, 517548, 283510, 96030

Offset: 1

Author

Alois P. Heinz, Jul 06 2015

Keywords

• nonn
• tabl

Comments

For the definition of Tesler matrices see A008608.

Sum_{k=1..n} k * T(n,k) = A259787(n).

Examples

			There are two 2 X 2 Tesler matrices: [1,0; 0,1], [0,1; 0,2], containing three 1's and one 2, thus row 2 gives [3, 1].
Triangle T(n,k) begins:
       1;
       3,      1;
      15,      5,     2;
     117,     37,    17,     7;
    1367,    418,   189,   100,    40;
   23329,   7027,  3058,  1688,   939,   357;
  570933, 171428, 72194, 39274, 24050, 13429, 4820;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..20, flattened

Crossrefs

Main diagonal gives A008608(n-1) for n>1.

Column k=1 gives A259843.

Row sums give A259842.

Cf. A259787.

Programs

Maple

g:= u-> `if`(u=0, 0, x^u):
b:= proc(n, i, l) option remember; (m->`if`(m=0, [1, g(n)], `if`(i=0,
     (p->p+[0, p[1]*g(n)])(b(l[1]+1, m-1, subsop(1=NULL, l))), add(
     (p->p+[0, p[1]*g(j)])(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=1..n))(b(1,n-1,[0$(n-1)])[2]):
seq(T(n), n=1..10);

Mathematica

g[u_] := If[u == 0, 0, x^u];
b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, {1, g[n]}, If[i == 0, # + {0, #[[1]] g[n]} & [b[l[[1]]+1, m-1, ReplacePart[l, 1 -> Nothing]]], Sum[# + {0, #[[1]] g[j]} & [b[n-j, i-1, ReplacePart[l, i -> l[[i]] + j]]], {j, 0, n}]]]][Length[l]];
T[n_] := Table[Coefficient[#, x, i], {i, 1, n}] & [b[1, n-1, Table[0, {n-1}]][[2]]];
Array[T, 10] // Flatten (* Jean-François Alcover, Oct 28 2020, after Maple *)