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

0,3

Links

Alois P. Heinz, Rows n = 0..250, flattened

Wikipedia, Partition of a set

Formula

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

Example

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;
  ...

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 *)

Crossrefs

Column k=0 gives A000085.

Row sums give A000110.

T(3n,n) gives A025035.

Cf. A048993, A124324, A124503, A288785.

Sequence in context: A114848 A135330 A135328 * A346419 A048941 A308300

Adjacent sequences: A355141 A355142 A355143 * A355145 A355146 A355147

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 20 2022

Status

approved