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A077611
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Number of adjacent pairs of form (odd,odd) among all permutations of {1,2,...,n}.
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5
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0, 0, 4, 12, 144, 720, 8640, 60480, 806400, 7257600, 108864000, 1197504000, 20118067200, 261534873600, 4881984307200, 73229764608000, 1506440871936000, 25609494822912000, 576213633515520000, 10948059036794880000, 267619220899430400000, 5620003638888038400000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| a(n) is also the number of permutations of [n+1] starting and ending by an even number. [From Olivier Gerard, Nov 7 2011].
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..400
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FORMULA
| a(n) = ceiling(n/2)*ceiling(n/2-1)*(n-1)!. Proof: There are ceiling(n/2) * ceiling(n/2-1) pairs (r, s) with r and s odd and distinct. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.
a(n) = (n-1)!*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8. - Bruno Berselli, Nov 07 2011
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EXAMPLE
| For n=4, the a(4) = 12 permutations of degree 5 starting and ending by an even number are : 21354, 21534, 23154, 23514, 25134, 25314, 41352, 41532, 43152, 43512, 45132, 45312.
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PROG
| (MAGMA) [Factorial(n-1)*(2*n*(n-1)-(2*n-1)*(-1)^n-1)/8 : n in [1..30]]; // Vincenzo Librandi, Nov 16 2011
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CROSSREFS
| Cf. A052618, A077612, A077613.
Sequence in context: A002029 A204321 A152121 * A052598 A032071 A173603
Adjacent sequences: A077608 A077609 A077610 * A077612 A077613 A077614
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KEYWORD
| nonn
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AUTHOR
| Leroy Quet Frank Ruskey (ruskey(AT)cs.uvic.ca) and Dean Hickerson (dean.hickerson(AT)yahoo.com), Nov 11 2002
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