A151359 Triangle read by rows: T(n,k) = number of partitions of [1..k] into n nonempty clumps of sizes 1, 2, 3, 4, 5 or 6 (n >= 0, 0 <= k <= 6n).

0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462, 0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856, 0, 0, 0, 0, 1, 10, 65, 350, 1701, 7770, 33985, 143605, 588511

Offset: 0

Author

N. J. A. Sloane, May 14 2009

Keywords

• nonn
• tabf

Comments

Row n has 6n+1 entries.

Examples

			Triangle begins:
[0, 1, 1, 1, 1, 1, 1]
[0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462]
[0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856]
[0, 0, 0, 0, 1, 10, 65, 350, 1701, 7770, 33985, 143605, 588511, 2341339, 9032023, 33668635, 120681561, 413104692, 1337944608, 4046710668, 11216721516, 27756632904, 58555088592, 96197645544, 96197645544]
[0, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42525, 246400, 1370985, 7383376, 38657619, 197212015, 980839860, 4752728981, 22399494117, 102410296989, 452572985865, 1924000439361, 7820764020069, 30157961878044, 109184327692440, 365935843649376, 1113006758944080, 2982608000091720, 6696799094545560, 11423951396577720, 11423951396577720]
...
		

Links

• Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
• David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Crossrefs

This is one of a sequence of triangles: A144331, A144385, A144643, A151338, A151359, ...

See A151511, A151512 for other versions.

Programs

Mathematica

Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; t[n, k_] := t[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n-1, j-1] a[j] t[n-j, k-1], {j, 0, n-k+1}]]; T[n_, k_] := t[k, n+1]; Table[Table[T[n, k], {k, 0, 6(n+1)} ], {n, 0, 4}] // Flatten (* Jean-François Alcover, Jan 20 2016, using Peter Luschny's Bell transform *)