A259787 -id:A259787 - cp's OEIS frontend

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

Alois P. Heinz, Jul 06 2015

For the definition of Tesler matrices see A008608.

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

			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;
  ...

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

Main diagonal gives A008608(n-1) for n>1.
Column k=1 gives A259843.
Row sums give A259842.
Cf. A259787.

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);

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 *)

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.

Original entry on oeis.org

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

Alois P. Heinz, Jul 05 2015

For the definition of Tesler matrices see A008608.

			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;
  ...

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

Row sums give A008608.

Cf. A000217, A259787.

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);

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 *)

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

cp's OEIS Frontend

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.

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