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Search: id:A008306
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| A008306 |
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Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)). |
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+0
11
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| 1, 2, 6, 3, 24, 20, 120, 130, 15, 720, 924, 210, 5040, 7308, 2380, 105, 40320, 64224, 26432, 2520, 362880, 623376, 303660, 44100, 945, 3628800, 6636960, 3678840, 705320, 34650, 39916800, 76998240, 47324376, 11098780, 866250, 10395
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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Also T(n,k) = number of derangements of {1..n} with k orbits.
Also T(n,k) = number of permutations of {1..n} with k cycles of length >= 2.
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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E.g.f. 1 + Sum_{1 <= 2k <= n} T(n, k)*t^n*u^k/n! = exp(-t*u)*(1-t)^(-u).
Recurrence: T(n, k) = (n-1)*(T(n-1, k) + T(n-2, k-1)) for 1<=k<=n/2 with boundary conditions T(0, 0)=1, T(n, 0)=0 for n>=1, T(n, k)=0 for k>n/2. - David Callan (callan(AT)stat.wisc.edu), May 16 2005
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EXAMPLE
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Rows 2 though 7 are:
1
2
6 3
24 20
120 130 15
720 924 210
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CROSSREFS
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See A106828 for another version. Diagonals give A000142, A000276, A000483. A079510 is the same triangle rearranged.
Sequence in context: A056195 A083169 A050125 this_sequence A144362 A125666 A111678
Adjacent sequences: A008303 A008304 A008305 this_sequence A008307 A008308 A008309
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KEYWORD
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tabf,nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001
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