A367468 Triangle read by rows: T(n,k) is the total number of movable letters in all members of the k-partitions of [n], with 1 <= k <= n.

0, 1, 0, 2, 4, 0, 3, 17, 9, 0, 4, 52, 68, 16, 0, 5, 139, 345, 190, 25, 0, 6, 346, 1474, 1440, 430, 36, 0, 7, 825, 5733, 8904, 4550, 847, 49, 0, 8, 1912, 21048, 49056, 38304, 11928, 1512, 64, 0, 9, 4343, 74385, 251250, 282135, 130998, 27342, 2508, 81, 0

Offset: 1

Author

Stefano Spezia, Nov 19 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  0;
  1,   0;
  2,   4,   0;
  3,  17,   9,   0;
  4,  52,  68,  16,  0;
  5, 139, 345, 190, 25, 0;
  ...
		

Links

• Toufik Mansour and Mark Shattuck, Counting set partitions by the number of movable letters, Journal of Difference Equations and Applications, 26:3, 384-403, (2020). On ResearchGate. See Theorem 8.

Crossrefs

Cf. A000290, A008277, A208658, A367469 (row sums).

Programs

Mathematica

 T[n_,k_]:=If[k==1,n-1,(2n-1)StirlingS2[n,k]/2-StirlingS2[n+1,k]/2+StirlingS2[n-1,k-2]/2]; Table[T[n,k],{n,10},{k,n}]//Flatten

Formula

T(n,1) = n-1 and T(n,k) = (2*n - 1)*S2(n,k)/2 - S2(n+1,k)/2 + S2(n-1,k-2)/2 for k > 1.

T(n,2) = A208658(n-1).

T(n,n-1) = A000290(n-1) for n > 1.