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

0,8

Links

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

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!).

Example

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;
  ...

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 *)

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.

Sequence in context: A309940 A081514 A109206 * A176506 A200596 A088422

Adjacent sequences: A292743 A292744 A292745 * A292747 A292748 A292749

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 22 2017

Extensions

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

Status

approved