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A145891
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k adjacent pairs of the form (odd,even) (0<=k<=floor(n/2)).
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3
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1, 1, 1, 1, 2, 4, 4, 16, 4, 12, 72, 36, 36, 324, 324, 36, 144, 1728, 2592, 576, 576, 9216, 20736, 9216, 576, 2880, 57600, 172800, 115200, 14400, 14400, 360000, 1440000, 1440000, 360000, 14400, 86400, 2592000, 12960000, 17280000, 6480000, 518400
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OFFSET
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0,5
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COMMENTS
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Also number of permutations of {1,2,...,n} having k adjacent pairs of the form (even,odd). Example: T(3,1) = 4 because we have 123, 213, 231 and 321.
Row n contains 1+floor(n/2) entries.
Sum of entries in row n = n! = A000142(n).
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LINKS
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FORMULA
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T(2n,k) = [n!*C(n,k)]^2; T(2n+1,k) = n!(n+1)! C(n,k) C(n+1,k).
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EXAMPLE
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T(3,1) = 4 because we have 123, 132, 312 and 321.
Triangle starts:
1;
1;
1, 1;
2, 4;
4, 16, 4;
12, 72, 36;
36, 324, 324, 36;
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MAPLE
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T:=proc(n, k) if `mod`(n, 2) = 0 then factorial((1/2)*n)^2*binomial((1/2)*n, k)^2 else factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2)*binomial((1/2)*n-1/2, k)*binomial((1/2)*n+1/2, k) end if end proc: for n from 0 to 11 do seq(T(n, k), k =0..floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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