A355144 Number T(n,k) of partitions of [n] having exactly k blocks of size at least three; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

1, 1, 2, 4, 1, 10, 5, 26, 26, 76, 117, 10, 232, 540, 105, 764, 2445, 931, 2620, 11338, 6909, 280, 9496, 53033, 48546, 4900, 35696, 253826, 324753, 64295, 140152, 1235115, 2131855, 691075, 15400, 568504, 6142878, 13792779, 6739876, 400400, 2390480, 31126539, 88890880, 61274213, 7217210

Offset: 0

Author

Alois P. Heinz, Jun 20 2022

Keywords

• nonn
• tabf

Examples

			T(4,1) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
T(6,2) = 10: 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Triangle T(n,k) begins:
       1;
       1;
       2;
       4,       1;
      10,       5;
      26,      26;
      76,     117,      10;
     232,     540,     105;
     764,    2445,     931;
    2620,   11338,    6909,    280;
    9496,   53033,   48546,   4900;
   35696,  253826,  324753,  64295;
  140152, 1235115, 2131855, 691075, 15400;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..250, flattened
• Wikipedia, Partition of a set

Crossrefs

Column k=0 gives A000085.

Row sums give A000110.

T(3n,n) gives A025035.

Cf. A048993, A124324, A124503, A288785.

Programs

Maple

b:= proc(n) option remember; expand(`if`(n=0, 1, add(
     `if`(i>2, x, 1)*binomial(n-1, i-1)*b(n-i), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jun 20 2022

Mathematica

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[If[i > 2, x, 1]*
     Binomial[n - 1, i - 1]*b[n - i], {i, 1, n}]]];
T[n_] := CoefficientList[b[n], x];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 25 2022, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A288785(n).