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A230797 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-3)/2)), read by rows.

%I A230797 #23 Dec 01 2017 17:41:24
%S A230797 1,1,2,6,24,104,16,528,192,3296,1472,272,23168,12800,4352,179712,
%T A230797 132352,42880,7936,1573632,1366016,530432,158720,15207424,14781952,
%U A230797 7662336,1911296,353792,158880768,178102272,101713920,31813632,8491008,1801996288,2282645504
%N A230797 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-3)/2)), read by rows.
%H A230797 Alois P. Heinz, <a href="/A230797/b230797.txt">Rows n = 0..170, flattened</a>
%e A230797 T(5,1) = 16: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231.
%e A230797 T(7,2) = 272: 1325476, 1326475, 1326574, ..., 6735241, 6745132, 6745231.
%e A230797 Triangle T(n,k) begins:
%e A230797 :  0 :       1;
%e A230797 :  1 :       1;
%e A230797 :  2 :       2;
%e A230797 :  3 :       6;
%e A230797 :  4 :      24;
%e A230797 :  5 :     104,      16;
%e A230797 :  6 :     528,     192;
%e A230797 :  7 :    3296,    1472,    272;
%e A230797 :  8 :   23168,   12800,   4352;
%e A230797 :  9 :  179712,  132352,  42880,   7936;
%e A230797 : 10 : 1573632, 1366016, 530432, 158720;
%p A230797 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
%p A230797       add(b(u-j, o+j-1, [1, 3, 1, 3][t])*`if`(t=4, x, 1), j=1..u)+
%p A230797       add(b(u+j-1, o-j, [2, 2, 4, 2][t]), j=1..o)))
%p A230797     end:
%p A230797 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
%p A230797 seq(T(n), n=0..15);  # _Alois P. Heinz_, Oct 30 2013
%t A230797 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1, 3}[[t]]]*If[t == 4, x, 1], {j, 1, u}] + Sum[b[u+j-1, o-j, {2, 2, 4, 2}[[t]]], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Oct 24 2016, after _Alois P. Heinz_ *)
%Y A230797 Columns k=0-2 give: A177520, A230832, A264077.
%Y A230797 T(2n-1,n-2) gives A000182(n) for n>=3.
%Y A230797 Row sums give: A000142.
%Y A230797 Cf. A242783, A242784, A295987.
%K A230797 nonn,tabf
%O A230797 0,3
%A A230797 _Alois P. Heinz_, Oct 30 2013