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Search: id:A059022
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| A059022 |
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Triangle of Stirling numbers of order 3. |
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+0
6
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| 1, 1, 1, 1, 10, 1, 35, 1, 91, 1, 210, 280, 1, 456, 2100, 1, 957, 10395, 1, 1969, 42735, 15400, 1, 4004, 158301, 200200, 1, 8086, 549549, 1611610, 1, 16263, 1827826, 10335325, 1401400, 1, 32631, 5903898, 57962905, 28028000, 1, 65382, 18682014
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OFFSET
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3,5
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COMMENT
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The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 3. This is the 3-associated Stirling number of the second kind (Comtet) or the Stirling number of order 3 (Fekete).
This is entered as a triangular array. The entries S_3(n,k) are zero for 3k>n, so these values are omitted. Initial entry in sequence is S_3(3,1).
Rows are of lengths 1,1,1,2,2,2,3,3,3,...
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
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FORMULA
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S_r(n+1, k)=k S_r(n, k)+binomial(n, r-1)S_r(n-r+1, k-1) for this sequence, r=3 G.f.: sum(S_r(n, k)u^k ((t^n)/(n!)), n=0..infty, k=0..infty)=exp(u(e^t-sum(t^i/i!, i=0..r-1)))
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EXAMPLE
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There are 10 ways of partitioning a set N of cardinality 6 into 2 blocks each of cardinality at least 3, so S_3(6,2)=10.
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CROSSREFS
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Cf. A008299, A059023, A059024, A059025.
Sequence in context: A050999 A070246 A085044 this_sequence A115097 A050313 A116574
Adjacent sequences: A059019 A059020 A059021 this_sequence A059023 A059024 A059025
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
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