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A152662
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} for which k is the maximal number of initial odd entries (0 <= k <= ceiling(n/2)).
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4
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1, 0, 1, 1, 1, 2, 2, 2, 12, 8, 4, 48, 36, 24, 12, 360, 216, 108, 36, 2160, 1440, 864, 432, 144, 20160, 11520, 5760, 2304, 576, 161280, 100800, 57600, 28800, 11520, 2880, 1814400, 1008000, 504000, 216000, 72000, 14400, 18144000, 10886400, 6048000, 3024000
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OFFSET
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0,6
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COMMENTS
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Sum of entries in row n is n! (A000142).
Row n has 1 + ceiling(n/2) entries.
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A152663(n).
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LINKS
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FORMULA
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T(2n+1,k) = n*k!*(2n-k)!*binomial(n+1,k) (n>= 1);
T(2n,k) = n*k!*(2n-1-k)!*binomial(n,k).
Sum_{k>=0} (k+1) * T(n,k) = A256881(n+1).
T(n,ceiling(n/2)) = A010551(n). (End)
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EXAMPLE
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T(3,0)=2 because we have 213 and 231.
T(4,2)=4 because we have 1324, 1342, 3124 and 3142.
Triangle starts:
1;
0, 1;
1, 1;
2, 2, 2;
12, 8, 4;
48, 36, 24, 12;
360, 216, 108, 36;
...
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MAPLE
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T := proc (n, k) if n=0 and k=0 then 1 elif n = 1 and k = 0 then 0 elif n = 1 and k = 1 then 1 elif `mod`(n, 2) = 1 then (1/2)*(n-1)*binomial((1/2)*n+1/2, k)*factorial(k)*factorial(n-1-k) else (1/2)*n*binomial((1/2)*n, k)*factorial(k)*factorial(n-1-k) end if end proc: for n from 0 to 11 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form
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MATHEMATICA
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T[n_, k_] := T[n, k] = Which[n == 0 && k == 0, 1, n == 1 && k == 1, 1, OddQ[n], (n - 1)/2*k!*(n - k - 1)!*Binomial[(n - 1)/2 + 1, k], True, n/2*k!*(n - k - 1)!*Binomial[n/2, k]];
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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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EXTENSIONS
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STATUS
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approved
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