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Search: id:A120095
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| A120095 |
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Triangle T(n,k) = total of number at last index for all set partitions of n into k parts. |
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4
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| 1, 1, 2, 1, 5, 3, 1, 11, 15, 4, 1, 23, 57, 34, 5, 1, 47, 195, 200, 65, 6, 1, 95, 633, 1010, 550, 111, 7, 1, 191, 1995, 4704, 3850, 1281, 175, 8, 1, 383, 6177, 20874, 24255, 11886, 2646, 260, 9, 1, 767, 18915, 89800, 143115, 97272, 31458, 4992, 369, 10
(list; table; graph; listen)
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OFFSET
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1,3
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FORMULA
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T(n,k) = k(k+1)/2 S2(n-1,k) + k S2(n-1,k-1) = 1/2 (S2(n+1,k) + S2(n,k) - S2(n-1,k-2)) = k T(n-1,k) + T(n-1,k-1) + S2(n-2,k-2), where S2 is the Stirling numbers of the second kind (A008277).
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EXAMPLE
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The set partitions of 4 objects into 2 parts are {1,1,1,2}, {1,1,2,1}, {1,1,2,2}, {1,2,1,1}, {1,2,1,2}, {1,2,2,1}, and {1,2,2,2}. The last terms of these sum to 2+1+2+1+2+1+2 = 11, so T(4,2) = 11.
Table starts:
1,
1,2,
1,5,3,
1,11,15,4,
1,23,57,34,5,
1,47,195,200,65,6
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CROSSREFS
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Cf. A008277, A120058. Row sums are A087648.
Adjacent sequences: A120092 A120093 A120094 this_sequence A120096 A120097 A120098
Sequence in context: A141483 A104731 A105728 this_sequence A130197 A106513 A054446
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KEYWORD
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nonn,tabl
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AUTHOR
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Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 07 2006
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