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%I #10 Jan 30 2017 21:22:51
%S 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,
%T 301,966,2989,8925,25641,70455,183183,441441,966966,1849848,2858856,
%U 2858856,0,0,0,0,1,10,65,350,1701,7770,33985,143605,588511
%N 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).
%C Row n has 6n+1 entries.
%H Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, <a href="http://arxiv.org/abs/1701.08394">Analysis of the Gift Exchange Problem</a>, arXiv:1701.08394, 2017.
%H David Applegate and N. J. A. Sloane, <a href="http://arxiv.org/abs/0907.0513">The Gift Exchange Problem</a>, arXiv:0907.0513 [math.CO], 2009.
%e Triangle begins:
%e [0, 1, 1, 1, 1, 1, 1]
%e [0, 0, 1, 3, 7, 15, 31, 63, 119, 210, 336, 462, 462]
%e [0, 0, 0, 1, 6, 25, 90, 301, 966, 2989, 8925, 25641, 70455, 183183, 441441, 966966, 1849848, 2858856, 2858856]
%e [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]
%e [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]
%e ...
%t 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 *)
%Y This is one of a sequence of triangles: A144331, A144385, A144643, A151338, A151359, ...
%Y See A151511, A151512 for other versions.
%K nonn,tabf
%O 0,11
%A _N. J. A. Sloane_, May 14 2009
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