A292746 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of n with exactly k kinds of 1's which are introduced in ascending order.

1, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 3, 8, 6, 1, 2, 5, 19, 26, 10, 1, 4, 7, 43, 97, 66, 15, 1, 4, 11, 93, 334, 361, 141, 21, 1, 7, 15, 197, 1095, 1778, 1066, 267, 28, 1, 8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1, 12, 30, 840, 10855, 36310, 43747, 23116, 5909, 751, 45, 1

Offset: 0

Author

Alois P. Heinz, Sep 22 2017

Keywords

• nonn
• tabl

Examples

			T(3,0) = 1: 3.
T(3,1) = 2: 21a, 1a1a1a.
T(3,2) = 3: 1a1a1b, 1a1b1a, 1a1b1b. (The two kinds of 1's are labeled 1a and 1b)
T(3,3) = 1: 1a1b1c.
Triangle T(n,k) begins:
  1;
  0,  1;
  1,  1,   1;
  1,  2,   3,    1;
  2,  3,   8,    6,    1;
  2,  5,  19,   26,   10,    1;
  4,  7,  43,   97,   66,   15,    1;
  4, 11,  93,  334,  361,  141,   21,   1;
  7, 15, 197, 1095, 1778, 1066,  267,  28,  1;
  8, 22, 409, 3482, 8207, 7108, 2668, 463, 36, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A002865, A000041(n-1) for n>0, A259401(n-2) for n>1, A320816, A320817, A320818, A320819, A320820, A320821, A320822, A320823.

Main diagonal and first lower diagonal give: A000012, A000217.

Row sums give A292503.

T(2n,n) gives A292747.

Cf. A292741, A292745.

Programs

Maple

f:= (n, k)-> add(Stirling2(n, j), j=0..k):
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      f(n, k), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      k^n, b(n, i-1, k) +b(n-i, min(i, n-i), k))
    end:
T:= (n, k)-> add((-1)^i*b(n$2, k-i)/((k-i)!*i!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..14);

Mathematica

f[n_, k_] := Sum[StirlingS2[n, j], {j, 0, k}];
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, f[n, k], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 17 2018, translated from Maple *)

Formula

T(n,k) = A292745(n,k) - A292745(n,k-1) for k>0. T(n,0) = A292745(n,0) = A002865(n).

T(n,k) = Sum_{i=0..k} (-1)^i * A292741(n, k-i) / ((k-i)!*i!).

Extensions

Definition clarified by N. J. A. Sloane, Dec 12 2020