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A138770
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} such that there are exactly k entries between the entries 1 and 2 (n>=2, 0<=k<=n-2).
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3
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2, 4, 2, 12, 8, 4, 48, 36, 24, 12, 240, 192, 144, 96, 48, 1440, 1200, 960, 720, 480, 240, 10080, 8640, 7200, 5760, 4320, 2880, 1440, 80640, 70560, 60480, 50400, 40320, 30240, 20160, 10080, 725760, 645120, 564480, 483840, 403200, 322560, 241920
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Sum of row n = n! = A000142(n).
T(n,0)=2(n-1)! = A052849(n-1)
T(n,1)=A052582(n-2).
T(n,2)=A052609(n-2).
T(n,3)=12*A005990(n-3).
T(n,4)=48*A061206(n-5).
T(n,n-2)=2(n-2)! (A052849).
Sum(k*T(n,k),k=0..n-2)=n!(n-2)/3=A090672(n-1).
The expected value of k is (n-2)/3 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Dec 19 2009]
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FORMULA
| T(n,k)=2*(n-k-1)(n-2)!
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EXAMPLE
| T(4,2)=4 because we have 1342, 1432, 2341 and 2431.
Triangle starts:
2;
4,2;
12,8,4;
48,36,24,12;
240,192,144,96,48;
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MAPLE
| T:=proc(n, k) if n-2 < k then 0 else (2*n-2*k-2)*factorial(n-2) end if end proc; for n from 2 to 10 do seq(T(n, k), k=0..n-2) end do; # yields sequence in triangular form
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MATHEMATICA
| Table[Table[2 (n - r) (n - 2)!, {r, 1, n - 1}], {n, 1, 10}] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Dec 19 2009]
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CROSSREFS
| Cf. A000142, A052489, A052582, A052609, A005990, A061206, A052849, A090672.
Sequence in context: A078034 A181091 A161795 * A006018 A152666 A153801
Adjacent sequences: A138767 A138768 A138769 * A138771 A138772 A138773
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 06 2008
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