A333156 -id:A333156 - cp's OEIS frontend

A166288 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k UDUD's (0<=k <= n-1; U=(1,1), D=(1,-1)).

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 4, 5, 6, 1, 1, 6, 12, 9, 8, 1, 1, 9, 23, 24, 14, 10, 1, 1, 17, 38, 56, 40, 20, 12, 1, 1, 26, 84, 100, 110, 60, 27, 14, 1, 1, 46, 145, 250, 210, 190, 84, 35, 16, 1, 1, 81, 280, 480, 580, 385, 301, 112, 44, 18, 1, 1, 135, 551, 995, 1225, 1155, 644, 448, 144, 54, 20, 1, 1

Offset: 1

Emeric Deutsch, Oct 12 2009

Sum of entries in row n = A004148(n+1) (the secondary structure numbers).
T(n,0) = A166289(n).
Sum(k*T(n,k), k=0..n-1) = A166290(n).

			T(5,2) = 6 because we have (UDUDUD)UUDD, UDU(UDUDUD)D, UUDD(UDUDUD), U(UDUD)D(UDUD), U(UDUDUD)DUD, and (UDUD)U(UDUD)D (the UDUD's are shown between parentheses).
Triangle starts:
  1;
  1,  1;
  2,  1,  1;
  2,  4,  1,  1;
  4,  5,  6,  1,  1;
  6, 12,  9,  8,  1, 1;
  9, 23, 24, 14, 10, 1, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A004148, A166289, A166290.

T(2n,n) gives A333156.

F := RootOf(z^3*G^2-(1+z-t*z)*(1-t*z-z^2)*G+(1+z-t*z)^2, G): Fser := series(F, z = 0, 15): for n to 12 do P[n] := sort(coeff(Fser, z, n)) end do: for n to 12 do seq(coeff(P[n], t, j), j = 0 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x or t=8, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, [2, 7, 4, 7, 2, 2, 8][t])
      +`if`(t=4, z, 1)  *b(x-1, y-1, [5, 3, 6, 3, 6, 8, 3][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=1..15);  # Alois P. Heinz, Jun 04 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x || t == 8, 0, If[x == 0, 1, Expand[b[x-1, y+1, {2, 7, 4, 7, 2, 2, 8}[[t]] ] + If[t == 4, z, 1]*b[x-1, y-1, {5, 3, 6, 3, 6, 8, 3}[[t]] ]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

G.f.: G(t,z) -1, where G=G(t,z) satisfies z^3*G^2 - (1+z-tz)(1-tz-z^2)G+(1+z-tz)^2=0.

A304361 Number of Dyck paths of semilength 2n having exactly n (possibly overlapping) occurrences of the consecutive pattern UDUD, where U=(1,1) and D=(1,-1).

Original entry on oeis.org

1, 1, 1, 9, 41, 244, 1555, 10037, 68599, 476981, 3399518, 24652718, 181411439, 1352123760, 10185964435, 77458698781, 593871350009, 4586247704944, 35646681303447, 278665636846853, 2189789189667782, 17288684906561300, 137081212514315262, 1091163063187762414

Offset: 0

Alois P. Heinz, May 11 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1072
Vaclav Kotesovec, Recurrence (of order 6)
Wikipedia, Counting lattice paths

Cf. A094507, A333156.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 2][t])
        +b(x-1, y-1, [1, 3, 1, 3][t])*`if`(t=4, z, 1))))
    end:
a:= n-> coeff(b(4*n, 0, 1), z, n):
seq(a(n), n=0..35);

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, 2}[[t]]] + b[x - 1, y - 1, {1, 3, 1, 3}[[t]]]*If[t == 4, z, 1]]]];
a[n_] := Coefficient[b[4*n, 0, 1], z, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

a(n) = A094507(2n,n).

a(n) ~ c * d^n / n^2, where d = 8.678461743575504549836851346229164298625429506253061911480810294... is the real root of equation 28*d^5 - 72*d^4 - 1119*d^3 - 3136*d^2 - 272*d - 16 = 0 and c = 0.15899091419445210968174633623072264522489566046427010886172717963... - Vaclav Kotesovec, Mar 25 2020

cp's OEIS Frontend

A166288 Triangle read by rows: T(n,k) is the number of Dyck paths with no UUU's and no DDD's, of semilength n and having k UDUD's (0<=k <= n-1; U=(1,1), D=(1,-1)).

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