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A162971
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Triangle read by rows: T(n,k) is number of non-derangement permutations of {1,2,...,n} having k cycles (1<=k<=n).
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1
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1, 0, 1, 0, 3, 1, 0, 8, 6, 1, 0, 30, 35, 10, 1, 0, 144, 210, 85, 15, 1, 0, 840, 1414, 735, 175, 21, 1, 0, 5760, 10752, 6664, 1960, 322, 28, 1, 0, 45360, 91692, 64764, 22449, 4536, 546, 36, 1, 0, 403200, 869040, 679580, 268380, 63273, 9450, 870, 45, 1, 0, 3991680
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Sum of entries in row n = A002467(n) (the number of non-derangement permutations of {1,2,...,n}.
T(n,2)=n*(n-2)! = A001048(n-1) for n>=3.
Sum(k*T(n,k), k=1..n) = A162972(n).
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FORMULA
| E.g.f.: G(t,z)=[1-exp(-tz)]/(1-z)^t.
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EXAMPLE
| T(4,2)=8 because we have (1)(234),(1)(243),(134)(2),(143)(2),(124)(3),(142)(3),(123)(4), and (132)(4).
Triangle starts:
1;
0,1;
0,3,1;
0,8,6,1;
0,30,35,10,1;
0,144,210,85,15,1.
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MAPLE
| G := (1-exp(-t*z))/(1-z)^t: Gser := simplify(series(G, z = 0, 15)): for n to 11 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form
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CROSSREFS
| A002467, A001048, A162972
Sequence in context: A090536 A187557 A052420 * A078521 A194938 A135871
Adjacent sequences: A162968 A162969 A162970 * A162972 A162973 A162974
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 22 2009
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