A384999 Irregular triangle read by rows: T(n,k) is the total number of partitions of all numbers <= n with k designated summands, n >= 0, 0 <= k <= A003056(n).

1, 1, 1, 1, 4, 1, 8, 1, 1, 15, 4, 1, 21, 13, 1, 33, 28, 1, 1, 41, 58, 4, 1, 56, 103, 13, 1, 69, 170, 35, 1, 87, 269, 77, 1, 1, 99, 404, 158, 4, 1, 127, 579, 298, 13, 1, 141, 810, 529, 35, 1, 165, 1116, 880, 86, 1, 189, 1470, 1431, 183, 1, 1, 220, 1935, 2214, 371, 4, 1, 238, 2475, 3348, 701, 13

Offset: 0

Author

Omar E. Pol, Jul 22 2025

Keywords

• nonn
• look
• tabf

Comments

When part i has multiplicity j > 0 exactly one part i is "designated".

The length of the row n is A002024(n+1) = 1 + A003056(n), hence the first positive element in column k is in the row A000217(k).

Column k gives the partial sums of the column k of A385001.

Columns converge to A210843 which is also the partial sums of A000716.

Examples

			Triangle begins:
---------------------------------------------
   n\k:   0    1     2      3     4    5   6
---------------------------------------------
   0 |    1;
   1 |    1,   1;
   2 |    1,   4;
   3 |    1,   8,    1;
   4 |    1,  15,    4;
   5 |    1,  21,   13;
   6 |    1,  33,   28,     1;
   7 |    1,  41,   58,     4;
   8 |    1,  56,  103,    13;
   9 |    1,  69,  170,    35;
  10 |    1,  87,  269,    77,    1;
  11 |    1,  99,  404,   158,    4;
  12 |    1, 127,  579,   298,   13;
  13 |    1, 141,  810,   529,   35;
  14 |    1, 165, 1116,   880,   86;
  15 |    1, 189, 1470,  1431,  183,   1;
  16 |    1, 220, 1935,  2214,  371,   4;
  17 |    1, 238, 2475,  3348,  701,  13;
  18 |    1, 277, 3156,  4894, 1269,  35;
  19 |    1, 297, 3921,  7036, 2187,  86;
  20 |    1, 339, 4866,  9871, 3639, 194;
  21 |    1, 371, 5906, 13629, 5872, 402,  1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..1000, flattened

Crossrefs

Columns: A000012 (k=0), A024916 (k=1).

Cf. A000217, A000716, A002024, A003056, A077285, A195825, A210843, A211970, A385001.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
    end:
g:= proc(n) option remember; `if`(n<0, 0, g(n-1)+b(n$2)) end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n)):
seq(T(n), n=0..20);  # Alois P. Heinz, Jul 22 2025