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A152667 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having k runs of even entries (n >= 2, 1 <= k <= floor(n/2)). For example, the permutation 321756498 has 3 runs of even entries: 2, 64 and 8. 3
2, 6, 12, 12, 48, 72, 144, 432, 144, 720, 2880, 1440, 2880, 17280, 17280, 2880, 17280, 129600, 172800, 43200, 86400, 864000, 1728000, 864000, 86400, 604800, 7257600, 18144000, 12096000, 1814400, 3628800, 54432000, 181440000, 181440000 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

Sum of entries in row n is n! (=A000142(n)). Row n contains floor(n/2) entries.

T(n,1) = A152873(n).

Sum_{k>=1} k*T(n,k) = A152668(n).

Mirror image of A145892.

LINKS

Table of n, a(n) for n=2..35.

FORMULA

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

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

EXAMPLE

T(4,2) = 12 because we have 1234, 3214, 1432, 3412, 2134, 2314 and their reverses.

Triangle starts:

    2;

    6;

   12,   12;

   48,   72;

  144,  432,  144;

  720, 2880, 1440;

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-1, k-1)*binomial(n+2, 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 .. floor((1/2)*n)) end do; # yields sequence in triangular form

CROSSREFS

Cf. A000142, A152666, A152668, A145892, A152873.

Sequence in context: A278256 A066791 A062723 * A145892 A216429 A250178

Adjacent sequences:  A152664 A152665 A152666 * A152668 A152669 A152670

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 14 2008

STATUS

approved

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Last modified February 25 05:15 EST 2020. Contains 332217 sequences. (Running on oeis4.)