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

1,2

Comments

For the definition of Tesler matrices see A008608.

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

Links

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

Example

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;
  ...

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 *)

Crossrefs

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

Column k=1 gives A259843.

Row sums give A259842.

Cf. A259787.

Sequence in context: A121335 A126454 A293558 * A228540 A144815 A065250

Adjacent sequences: A259838 A259839 A259840 * A259842 A259843 A259844

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 06 2015

Status

approved