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A161133
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Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having exactly k odd fixed points (0<=k<=ceil(n/2)).
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1
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1, 0, 1, 1, 1, 3, 2, 1, 14, 8, 2, 64, 42, 12, 2, 426, 234, 54, 6, 2790, 1704, 468, 72, 6, 24024, 12864, 3024, 384, 24, 205056, 120120, 32160, 5040, 480, 24, 2170680, 1145400, 272400, 37200, 3000, 120, 22852200, 13024080, 3436200, 544800, 55800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Row n contains 1 + ceil(n/2) entries.
Sum of row n is n! = A000142(n).
T(n,0)=A161131(n).
Sum(k*T(n,k), k>=0) = A052558(n-1).
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FORMULA
| T(n,k)=binom(ceil(n/2), k)*Sum[(-1)^j*(n-k-j)!*binom(ceil(n/2)-k, j), j=0..ceil(n/2)-k].
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EXAMPLE
| T(3,0)=3 because we have 312, 231, 321; T(3,2)=1 because we have 123.
Triangle starts:
1;
0,1;
1,1;
3,2,1;
14,8,2;
64,42,12,2;
426,234,54,6.
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MAPLE
| T := proc (n, k) options operator, arrow: binomial(ceil((1/2)*n), k)*add((-1)^j*binomial(ceil((1/2)*n)-k, j)*factorial(n-k-j), j = 0 .. ceil((1/2)*n)-k) end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
| A000142, A161131, A052558, A161134
Sequence in context: A129652 A154921 A127126 * A112911 A152405 A152400
Adjacent sequences: A161130 A161131 A161132 * A161134 A161135 A161136
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 18 2009
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