A184176 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having k adjacent blocks of size 3, i.e., blocks of the form (i,i+1,i+2) (0 <= k <= floor(n/3)).

1, 1, 2, 4, 1, 13, 2, 46, 6, 184, 18, 1, 805, 69, 3, 3840, 288, 12, 19775, 1324, 47, 1, 109180, 6578, 213, 4, 642382, 35136, 1032, 20, 4007712, 200398, 5390, 96, 1, 26399764, 1214136, 30027, 505, 5, 182939900, 7778856, 177744, 2792, 30, 1329327991, 52501052, 1112969, 16362, 170, 1

Offset: 0

Author

Emeric Deutsch, Feb 09 2011

Keywords

• nonn
• tabf

Comments

Number of entries in row n is 1 + floor(n/3).

Sum of entries in row n = A000110(n) (the Bell numbers).

T(n,0) = A184177(n).

Sum_{k>=0}k*T(n,k) = A052889(n-2).

Examples

			T(4,1) = 2 because we have 123-4 and 1-234.
Triangle starts:
    1;
    1;
    2;
    4,  1;
   13,  2;
   46,  6;
  184, 18,  1;
		

Links

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

Crossrefs

Cf. A000110, A052889, A184174, A184175, A184177.

Programs

Maple

with(combinat): q := 3: a := proc (n, k) options operator, arrow: sum((-1)^(k+j)*binomial(j, k)*binomial(n+j-j*q, j)*bell(n-j*q), j = k .. floor(n/q)) end proc: for n from 0 to 15 do seq(a(n, k), k = 0 .. floor(n/q)) end do; # yields sequence in triangular form

Mathematica

q = 3; a[n_, k_] := Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[n+j-j*q, j]* BellB[n-j*q], {j, k, Floor[n/q]}]; Table[a[n, k], {n, 0, 15}, {k, 0, Floor[n/q]}] // Flatten (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

Formula

T(n,k) = Sum_{j=k..floor(n/3)}(-1)^(k+j) * C(j,k) * C(n-2j,j) * Bell(n-3j).