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%I #16 Apr 28 2020 07:33:25
%S 38,587,4785,31398,190050,1043248,5324534,25711105,119092876,
%T 533680433,2329450085,9955122396,41824314441,173289259905,
%U 709861015186,2880803895035,11601285215222,46422795985447,184784743066842,732324944072523,2891815190097065,11385122145001833
%N Number of Dyck paths of semilength n such that all sixteen consecutive step patterns of length 4 occur at least once.
%H Alois P. Heinz, <a href="/A243820/b243820.txt">Table of n, a(n) for n = 10..70</a>
%e a(10) = 38: 10101100111101000010, 10101101001111000010, 10101111000011010010, 10101111001101000010, 10101111010000110010, 10101111010011000010, 10110011110100001010, 10110011110101000010, 10110100111100001010, 10110101001111000010, 10111100001101001010, 10111100001101010010, 10111100110100001010, 10111100110101000010, 10111101000011001010, 10111101001100001010, 10111101010000110010, 10111101010011000010, 11001011110000110100, 11001011110100001100, 11001101011110000100, 11001101111000010100, 11001111000010110100, 11001111010000101100, 11001111010110000100, 11001111011000010100, 11010010111100001100, 11010011110000101100, 11010110011110000100, 11010111100001001100, 11010111100001100100, 11010111100100001100, 11010111100110000100, 11011001111000010100, 11011110000101001100, 11011110000110010100, 11011110010100001100, 11011110011000010100. Here 1=Up=(1,1), 0=Down=(1,-1).
%p b:= proc(x, y, t, s) option remember; `if`(y<0 or y>x, 0,
%p `if`(x=0, `if`(s={}, 1, 0), `if`(nops(s)>x, 0, add(
%p b(x-1, y-1+2*j, irem(2*t+j, 8), s minus {2*t+j}), j=0..1))))
%p end:
%p a:= n-> add(b(2*n-3, l[], {$0..15}), l=[[1, 5], [1, 6], [3, 7]]):
%p seq(a(n), n=10..20);
%t b[x_, y_, t_, s_List] := b[x, y, t, s] = If[y < 0 || y > x, 0, If[x == 0, If[s == {}, 1, 0], If[Length[s] > x, 0, Sum[b[x - 1, y - 1 + 2 j, Mod[2 t + j, 8], s ~Complement~ {2 t + j}], {j, 0, 1}]]]];
%t a[n_] := Sum[b[2 n - 3, Sequence @@ l, Range[0, 15]], {l, {{1, 5}, {1, 6}, {3, 7}}}];
%t a /@ Range[10, 31] (* _Jean-François Alcover_, Apr 28 2020, after _Alois P. Heinz_ *)
%Y Cf. A242257, A243882, A243965, A243966.
%K nonn
%O 10,1
%A _Alois P. Heinz_, Jun 11 2014
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