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A008291
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Triangle of rencontres numbers.
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14
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1, 2, 3, 9, 8, 6, 44, 45, 20, 10, 265, 264, 135, 40, 15, 1854, 1855, 924, 315, 70, 21, 14833, 14832, 7420, 2464, 630, 112, 28, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 14684570
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OFFSET
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2,2
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COMMENTS
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T(n,k) = number of permutations of n elements with k fixed points.
T(n,n-1)=0 and T(n,n)=1 are ommitted from the array. -Geoffrey Critzer, Nov 28 2011.
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
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LINKS
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T. D. Noe, Rows n=2..50, flattened
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FORMULA
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E.g.f. for column k: (x^k/k!)(exp(-x)/(1-x)). - Geoffrey Critzer, Nov 28 2011
Row generating polynomials appear to be given by -1 + sum {k = 0..n} (-1)^(n+k)*C(n,k)*(1+k*x)^(n-k)*(2+(k-1)*x)^k. - Peter Bala, Dec 29 2011
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EXAMPLE
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Triangle begins:
1
2 3
9 8 6
44 45 20 10
265 264 135 40 15
1854 1855 924 315 70 21
14833 14832 7420 2464 630 112 28
133496 133497 66744 22260 5544 1134 168 36
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MAPLE
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T:= proc(n, k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
(T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
end:
seq(seq(T(n, k), k=0..n-2), n=2..12); # Alois P. Heinz, Mar 17 2013
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MATHEMATICA
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Prepend[Flatten[f[list_]:=Select[list, #>1&]; Map[f, Drop[Transpose[Table[d = Exp[-x]/(1 - x); Range[0, 10]! CoefficientList[Series[d x^k/k!, {x, 0, 10}], x], {k, 0, 8}]], 3]]], 1] (* Geoffrey Critzer, Nov 28 2011 *)
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PROG
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(PARI) {T(n, k)= if(k<0|k>n, 0, n!/k!*sum(i=0, n-k, (-1)^i/i!))}
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CROSSREFS
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T(n,k) = binomial(n,k)*A000166(n-k). Cf. A008290.
Diagonals give A000217, A007290, A060008, A060836, A000166, A000240, A000387, A000449, A000475.
Sequence in context: A021421 A152812 A086565 * A122665 A133066 A131988
Adjacent sequences: A008288 A008289 A008290 * A008292 A008293 A008294
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KEYWORD
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nonn,tabl,nice,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Comments and more terms from Michael Somos, Apr 26 2000.
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STATUS
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approved
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