A365676 Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct part sizes, for 0 <= k <= n.

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0, 0, 2, 5, 0, 0, 0, 0, 4, 6, 1, 0, 0, 0, 0, 2, 11, 2, 0, 0, 0, 0, 0, 4, 13, 5, 0, 0, 0, 0, 0, 0, 3, 17, 10, 0, 0, 0, 0, 0, 0, 0, 4, 22, 15, 1, 0, 0, 0, 0, 0, 0, 0, 2, 27, 25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 6, 29, 37, 5, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Peter Luschny, Sep 15 2023

Keywords

• nonn
• tabl

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 2,  0;
  [3] 0, 2,  1,  0;
  [4] 0, 3,  2,  0, 0;
  [5] 0, 2,  5,  0, 0, 0;
  [6] 0, 4,  6,  1, 0, 0, 0;
  [7] 0, 2, 11,  2, 0, 0, 0, 0;
  [8] 0, 4, 13,  5, 0, 0, 0, 0, 0;
  [9] 0, 3, 17, 10, 0, 0, 0, 0, 0, 0;
		

Links

• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75-113; Coll. Papers II, pp. 303-341.

Crossrefs

Variants: A116608 (nonzero terms), A060177.

Cf. A000041 (row sums), A000005 (T(n,1)), A002133 (T(n,2)), A002134 (T(n,3)), A365630 (T(n,4)), A365631 (T(n,5)).

Programs

Maple

P := proc(n, k, r) option remember; local j;  # after Amir Livne Bar-on
  if n = 0 then return ifelse(k = 0, 1, 0) fi;
  if k = 0 or r = 0 then return 0 fi;
  add(P(n - r * j, k - 1, r - 1), j = 1..iquo(n, r)) + P(n, k, r - 1) end:
A365676row := n -> local k; seq(P(n, k, n), k = 0..n):
seq(print(A365676row(n)), n = 0..9);
# Using the generating function:
p := product(1 + t*x^j/(1 - x^j), j = 1..20):
ser := series(p, x, 20):
seq(seq(coeff(coeff(ser, x, n), t, k), k = 0..n), n = 0..9);

Mathematica

P[n_, k_, r_] := P[n, k, r] = Which[n == 0, If[k == 0, 1, 0], k == 0 || r == 0, 0, True, Sum[P[n-r*j, k-1, r-1], {j, 1, Quotient[n, r]}]+P[n, k, r-1]]; A365676row[n_] := Table[P[n, k, n], {k, 0, n}]; Table[A365676row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 21 2023, from 1st Maple program *)

PARI

T(n,k) = my(nb=0); forpart(p=n, if (#Set(p) == k, nb++)); nb; \\ Michel Marcus, Sep 17 2023

Python

# after Amir Livne Bar-on
from functools import cache
@cache
def P(n: int, k: int, r: int) -> int:
    if n == 0: return 1 if k == 0 else 0
    if k == 0 or r == 0: return 0
    return sum(P(n - r * j, k - 1, r - 1)
               for j in range(1, n // r + 1)) + P(n, k, r - 1)
def A365676Row(n) -> list[int]:
    return [P(n, k, n) for k in range(n + 1)]
for n in range(10): print(A365676Row(n))

Formula

T(n, k) = [t^k][x^n] Product_{j>=1} (1 + t*x^j / (1 - x^j)).

T(n, k) = 0 for n>0 and k=0. T(n,k) = 0 for k > floor([sqrt(1+8n)-1]/2). - Chai Wah Wu, Sep 15 2023