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A232187 Number T(n,k) of parity alternating permutations of [n] with exactly k descents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows. 2
1, 1, 2, 1, 1, 5, 3, 2, 8, 2, 20, 44, 8, 6, 66, 66, 6, 114, 594, 414, 30, 24, 624, 1584, 624, 24, 864, 8784, 14544, 4464, 144, 120, 6840, 36240, 36240, 6840, 120, 8280, 147720, 471120, 353520, 55320, 840, 720, 86400, 857520, 1739520, 857520, 86400, 720, 96480 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

T(2n+2,n) = T(2n+1,n) + T(2n+3,n+1).

LINKS

Alois P. Heinz, Rows n = 0..29, flattened

FORMULA

T(2n+1,k) = n! * A173018(n+1,k) = A000142(n) * A173018(n+1,k).

EXAMPLE

T(5,0) = 2: 12345, 34125.

T(5,1) = 8: 12543, 14325, 14523, 32145, 34521, 52143, 52341, 54123.

T(5,2) = 2: 32541, 54321.

T(6,2) = 8: 163254, 165432, 321654, 325416, 541632, 543216, 632541, 654321.

T(7,0) = 6: 1234567, 1256347, 3412567, 3456127, 5612347, 5634127.

T(7,1) = 66: 1234765, 1236547, 1236745, ..., 7456123, 7612345, 7634125.

T(7,2) = 66: 1254763, 1276543, 1432765, ..., 7652143, 7652341, 7654123.

T(7,3) = 6: 3254761, 3276541, 5432761, 5476321, 7632541, 7654321.

Triangle T(n,k) begins:

:  0 :   1;

:  1 :   1;

:  2 :   2;

:  3 :   1,    1;

:  4 :   5,    3;

:  5 :   2,    8,     2;

:  6 :  20,   44,     8;

:  7 :   6,   66,    66,     6;

:  8 : 114,  594,   414,    30;

:  9 :  24,  624,  1584,   624,   24;

: 10 : 864, 8784, 14544,  4464,  144;

: 11 : 120, 6840, 36240, 36240, 6840, 120;

CROSSREFS

Column k=0 gives: A199660.

Row sums give: A092186 (for n>0).

T(2n+1,n) = A000142(n).

T(2n+2,n) = A001048(n+1).

Sequence in context: A263296 A259862 A182930 * A076241 A316399 A139347

Adjacent sequences:  A232184 A232185 A232186 * A232188 A232189 A232190

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Nov 20 2013

STATUS

approved

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Last modified February 24 18:55 EST 2021. Contains 341584 sequences. (Running on oeis4.)