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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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13
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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; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| 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.
a(n) is related to the subfactorials A000166(n). The row sums of the n'th row = !n...1=1, 2=2, 6+3=9, 24+20=44, 120+130+15=265.... [From Gary Detlefs (gdetlefs(AT)aol.com), Jul 14 2010]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.
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LINKS
| Eric Weisstein's World of Mathematics, Permutation Cycle
Eric Weisstein's World of Mathematics, Stirling Number of the First Kind
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FORMULA
| 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
E.g.f. for column k: B(A(x)) where A(x)= Log(1/1-x)-x and B(x)=x^k/k!
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EXAMPLE
| Rows 2 though 7 are:
1
2
6 3
24 20
120 130 15
720 924 210
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MAPLE
| A008306 := proc(n, k) local j;
add(binomial(j, n-2*k)*A008517(n-k, j), j=0..n-k) end;
seq(print(seq(A008306(n, k), k=1..iquo(n, 2))), n=2..12):
- Peter Luschny, Apr 20 2011
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MATHEMATICA
| t[0, 0] = 1; t[n_, 0] = 0; t[n_, k_] /; k > n/2 = 0; t[n_, k_] := t[n, k] = (n - 1)*(t[n - 1, k] + t[n - 2, k - 1]); A008306 = Flatten[ Table[ t[n, k], {n, 2, 12}, {k, 1, Quotient[n, 2]}]] (* From Jean-François Alcover, Jan 25 2012, after David Callan *)
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CROSSREFS
| See A106828 for another version. Diagonals give A000142, A000276, A000483. A079510 is the same triangle rearranged.
Sequence in context: A083169 A050125 A178667 * A144362 A125666 A172031
Adjacent sequences: A008303 A008304 A008305 * A008307 A008308 A008309
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KEYWORD
| tabf,nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001
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