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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.

%I A232187 #16 Nov 25 2013 14:22:35
%S A232187 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,
%T A232187 24,864,8784,14544,4464,144,120,6840,36240,36240,6840,120,8280,147720,
%U A232187 471120,353520,55320,840,720,86400,857520,1739520,857520,86400,720,96480
%N 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.
%C A232187 T(2n+1,k) = T(2n+1,n-k).
%C A232187 T(2n+2,n) = T(2n+1,n) + T(2n+3,n+1).
%H A232187 Alois P. Heinz, <a href="/A232187/b232187.txt">Rows n = 0..29, flattened</a>
%F A232187 T(2n+1,k) = n! * A173018(n+1,k) = A000142(n) * A173018(n+1,k).
%e A232187 T(5,0) = 2: 12345, 34125.
%e A232187 T(5,1) = 8: 12543, 14325, 14523, 32145, 34521, 52143, 52341, 54123.
%e A232187 T(5,2) = 2: 32541, 54321.
%e A232187 T(6,2) = 8: 163254, 165432, 321654, 325416, 541632, 543216, 632541, 654321.
%e A232187 T(7,0) = 6: 1234567, 1256347, 3412567, 3456127, 5612347, 5634127.
%e A232187 T(7,1) = 66: 1234765, 1236547, 1236745, ..., 7456123, 7612345, 7634125.
%e A232187 T(7,2) = 66: 1254763, 1276543, 1432765, ..., 7652143, 7652341, 7654123.
%e A232187 T(7,3) = 6: 3254761, 3276541, 5432761, 5476321, 7632541, 7654321.
%e A232187 Triangle T(n,k) begins:
%e A232187 :  0 :   1;
%e A232187 :  1 :   1;
%e A232187 :  2 :   2;
%e A232187 :  3 :   1,    1;
%e A232187 :  4 :   5,    3;
%e A232187 :  5 :   2,    8,     2;
%e A232187 :  6 :  20,   44,     8;
%e A232187 :  7 :   6,   66,    66,     6;
%e A232187 :  8 : 114,  594,   414,    30;
%e A232187 :  9 :  24,  624,  1584,   624,   24;
%e A232187 : 10 : 864, 8784, 14544,  4464,  144;
%e A232187 : 11 : 120, 6840, 36240, 36240, 6840, 120;
%Y A232187 Column k=0 gives: A199660.
%Y A232187 Row sums give: A092186 (for n>0).
%Y A232187 T(2n+1,n) = A000142(n).
%Y A232187 T(2n+2,n) = A001048(n+1).
%K A232187 nonn,tabf
%O A232187 0,3
%A A232187 _Alois P. Heinz_, Nov 20 2013