A384999 - OEIS

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

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OFFSET

0,5

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.

LINKS

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

EXAMPLE

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;

...

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

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Sum[Expand[b[n-i*j, i-1]*j*x], {j, 1, n/i}]]];

g[n_] := g[n] = If[n < 0, 0, g[n-1] + b[n, n]];

T[n_] := CoefficientList[g[n], x];

Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 04 2025, after Alois P. Heinz *)

CROSSREFS

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

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

Sequence in context: A084884 A329998 A143320 * A089146 A297402 A239666

Adjacent sequences: A384996 A384997 A384998 * A385000 A385001 A385002

KEYWORD

nonn,look,tabf

AUTHOR

Omar E. Pol, Jul 22 2025

STATUS

approved