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A152666 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of odd entries (1<=k<=ceiling(n/2)). For example, the permutation 321756498 has 3 runs of odd entries: 3, 175 and 9. 3
1, 2, 4, 2, 12, 12, 36, 72, 12, 144, 432, 144, 576, 2592, 1728, 144, 2880, 17280, 17280, 2880, 14400, 115200, 172800, 57600, 2880, 86400, 864000, 1728000, 864000, 86400, 518400, 6480000, 17280000, 12960000, 2592000, 86400, 3628800, 54432000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sum of entries in row n is n! (=A000142(n)).

Row n contains ceiling(n/2) entries.

T(n,1) = A010551(n+1).

Sum_{k>=1} k*T(n,k) = A052618(n-1).

Mirror image of A134435.

LINKS

Table of n, a(n) for n=1..38.

FORMULA

T(2n,k) = (n!)^2*binomial(n+1,k)*binomial(n-1,k-1).

T(2n+1,k) = n!*(n+1)!*binomial(n,k-1)*binomial(n+1,k).

EXAMPLE

T(3,2)=2 because we have 123 and 321.

T(4,2)=12 because we have 1234, 1432, 3214, 3412, 1243, 3241 and their reverses.

Triangle starts:

1;

2;

4,2;

12,12;

36,72,12;

144,432,144;

576,2592,1728,144.

MAPLE

ae := proc (n, k) options operator, arrow: factorial(n)^2*binomial(n+1, k)*binomial(n-1, k-1) end proc: ao := proc (n, k) options operator, arrow: factorial(n)*factorial(n+1)*binomial(n, k-1)*binomial(n+1, k) end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 1 .. ceil((1/2)*n)) end do; # yields sequence in triangular form

MATHEMATICA

T[n_?EvenQ, k_] := (n/2)!^2*Binomial[n/2 - 1, k - 1]*Binomial[n/2 + 1, k]; T[n_?OddQ, k_] := ((n - 1)/2 + 1)!*((n - 1)/2)!*Binomial[(n - 1)/2 + 1, k]*Binomial[(n - 1)/2, k - 1]; Table[T[n, k], {n, 1, 12}, {k, 1, Floor[(n + 1)/2]}] // Flatten (* Jean-Fran├žois Alcover, Nov 13 2016 *)

CROSSREFS

Cf. A000142, A010551, A052618, A152667, A134435.

Sequence in context: A161795 A138770 A006018 * A153801 A062867 A264027

Adjacent sequences:  A152663 A152664 A152665 * A152667 A152668 A152669

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 14 2008

STATUS

approved

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Last modified November 20 02:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)