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A243366 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.

%I A243366 #21 Mar 27 2021 15:15:05
%S A243366 1,1,2,5,13,1,37,5,112,19,1,352,70,7,1136,259,34,1,3742,962,149,9,
%T A243366 12529,3585,627,54,1,42513,13399,2584,279,11,145868,50201,10529,1334,
%U A243366 79,1,505234,188481,42606,6092,474,13,1764157,709001,171563,27048,2561,109,1
%N A243366 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.
%C A243366 Conjecture: Generally, column k is asymptotic to c(k) * d^n * n^(k-3/2), where d = 3.8821590268628506747194368909643384... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c(k) are specific constants (independent on n). - _Vaclav Kotesovec_, Jun 05 2014
%H A243366 Alois P. Heinz, <a href="/A243366/b243366.txt">Rows n = 0..200, flattened</a>
%e A243366 T(4,1) = 1: UDUUDUDD.
%e A243366 T(5,1) = 5: UDUDUUDUDD, UDUUDUDDUD, UDUUDUDUDD, UDUUDUUDDD, UUDUUDUDDD.
%e A243366 T(6,1) = 19: UDUDUDUUDUDD, UDUDUUDUDDUD, UDUDUUDUDUDD, UDUDUUDUUDDD, UDUUDUDDUDUD, UDUUDUDDUUDD, UDUUDUDUDDUD, UDUUDUDUDUDD, UDUUDUDUUDDD, UDUUDUUDDDUD, UDUUDUUDDUDD, UDUUDUUUDDDD, UUDDUDUUDUDD, UUDUDUUDUDDD, UUDUUDUDDDUD, UUDUUDUDDUDD, UUDUUDUDUDDD, UUDUUDUUDDDD, UUUDUUDUDDDD.
%e A243366 T(6,2) = 1: UDUUDUUDUDDD.
%e A243366 T(7,2) = 7: UDUDUUDUUDUDDD, UDUUDUDUUDUDDD, UDUUDUUDUDDDUD, UDUUDUUDUDDUDD, UDUUDUUDUDUDDD, UDUUDUUDUUDDDD, UUDUUDUUDUDDDD.
%e A243366 T(8,3) = 1: UDUUDUUDUUDUDDDD.
%e A243366 Triangle T(n,k) begins:
%e A243366 :  0 :     1;
%e A243366 :  1 :     1;
%e A243366 :  2 :     2;
%e A243366 :  3 :     5;
%e A243366 :  4 :    13,    1;
%e A243366 :  5 :    37,    5;
%e A243366 :  6 :   112,   19,   1;
%e A243366 :  7 :   352,   70,   7;
%e A243366 :  8 :  1136,  259,  34,  1;
%e A243366 :  9 :  3742,  962, 149,  9;
%e A243366 : 10 : 12529, 3585, 627, 54, 1;
%p A243366 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
%p A243366      `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 5, 2, 4][t])*
%p A243366      `if`(t=6, z, 1) +b(x-1, y-1, [1, 3, 1, 3, 6, 1][t]))))
%p A243366     end:
%p A243366 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
%p A243366 seq(T(n), n=0..20);
%t A243366 b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 2, 4}[[t]]]*If[t == 6, z, 1] + b[x-1, y-1, {1, 3, 1, 3, 6, 1}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Feb 05 2015, after _Alois P. Heinz_ *)
%Y A243366 Column k=0-10 give: A243412, A243413, A243414, A243415, A243416, A243417, A243418, A243419, A243420, A243421, A243422.
%Y A243366 Row sums give A000108.
%Y A243366 T(n,floor(n/2)-1) gives A093178(n) for n>3.
%Y A243366 T(45,k) = A243752(45,k).
%Y A243366 T(n,0) = A243753(n,45).
%K A243366 nonn,tabf
%O A243366 0,3
%A A243366 _Alois P. Heinz_, Jun 03 2014