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A231777 Number T(n,k) of permutations of [n] with exactly k ascents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

%I A231777 #10 Nov 14 2013 15:17:56
%S A231777 1,1,1,1,4,2,8,14,2,54,60,6,162,402,150,6,1536,2712,768,24,6144,19704,
%T A231777 12744,1704,24,75000,183120,94320,10320,120,375000,1473720,1392720,
%U A231777 365520,21720,120,5598720,17522640,13631040,3011040,152640,720,33592320,156250800
%N A231777 Number T(n,k) of permutations of [n] with exactly k ascents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.
%H A231777 Alois P. Heinz, <a href="/A231777/b231777.txt">Rows n = 0..23, flattened</a>
%e A231777 T(4,0) = 8: 1324, 2413, 2431, 3241, 4132, 4213, 4231, 4321.
%e A231777 T(4,1) = 14: 1243, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 3124, 3142, 3214, 3421, 4123, 4312.
%e A231777 T(4,2) = 2: 1234, 3412.
%e A231777 T(5,2) = 6: 12345, 12534, 34125, 34512, 51234, 53412.
%e A231777 T(6,3) = 6: 123456, 125634, 341256, 345612, 561234, 563412.
%e A231777 Triangle T(n,k) begins:
%e A231777 :  0 :      1;
%e A231777 :  1 :      1;
%e A231777 :  2 :      1,       1;
%e A231777 :  3 :      4,       2;
%e A231777 :  4 :      8,      14,       2;
%e A231777 :  5 :     54,      60,       6;
%e A231777 :  6 :    162,     402,     150,      6;
%e A231777 :  7 :   1536,    2712,     768,     24;
%e A231777 :  8 :   6144,   19704,   12744,   1704,    24;
%e A231777 :  9 :  75000,  183120,   94320,  10320,   120;
%e A231777 : 10 : 375000, 1473720, 1392720, 365520, 21720, 120;
%Y A231777 Column k=0 gives: A231601.
%Y A231777 Row sums and T(2n,n) give: A000142.
%Y A231777 T(n,floor(n/2)) gives: A081123(n+1).
%Y A231777 Cf. A004526.
%K A231777 nonn,tabf
%O A231777 0,5
%A A231777 _Alois P. Heinz_, Nov 13 2013