A243828 -id:A243828 - cp's OEIS frontend

A243836 Number A(n,k) of Dyck paths of semilength n having exactly ten (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16796, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1210, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 66, 15730, 0, 0

Offset: 0

Alois P. Heinz, Jun 11 2014

			Square array A(n,k) begins:
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  16796, 16796,  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
      0,     0, 55, 1, 0, 1, 0, 0, 0, 0, 1, 0, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Main diagonal gives A243779 or column k=10 of A243752.

Cf. A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835.

A243771 Number of Dyck paths of semilength n having exactly two (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).

Original entry on oeis.org

1, 1, 2, 12, 69, 98, 180, 1056, 3967, 18357, 77685, 264563, 1245762, 1915056, 5303208, 24548040, 107835695, 375494210, 1898502240, 4942470942, 23489565822, 104559681798, 413327570240, 1426320927138, 6025235528016, 19911812844324, 87316285518504

Offset: 2

Alois P. Heinz, Jun 10 2014

nonn

			a(2) = 1: (UD)[UD].
a(3) = 1: (U[U)U]DDD.
a(4) = 2: U(UDD)U[UDD], UU(UDD)[UDD].
a(5) = 12: (UD[U)DU]UDDUD, (UD[U)DU]UUDDD, (UDU)UDD[UDU]D, (UDU)[UDU]DDUD, (UDU)[UDU]UDDD, (UDU)U[UDU]DDD, UUDD(UD[U)DU]D, U(UDU)DD[UDU]D, U(UD[U)DU]DDUD, U(UD[U)DU]UDDD, U(UDU)[UDU]DDD, UU(UD[U)DU]DDD.

Alois P. Heinz, Table of n, a(n) for n = 2..350

Column k=2 of A243752.

Main diagonal of A243828.

A243872 Number of Dyck paths of semilength n having exactly 2 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 4, 16, 65, 263, 1077, 4419, 18132, 74368, 304778, 1247972, 5105477, 20867862, 85219608, 347724794, 1417697157, 5775652743, 23512922998, 95657223246, 388912046916, 1580241458120, 6417249216667, 26046042351889, 105661066012240, 428430870576913

Offset: 9

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..400
Vaclav Kotesovec, Recurrence (of order 14)

Column k=2 of A243881.

Column k=738 of A243828.

b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
     series(b(x-1, y+1, [2, 2, 4, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
      z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 1, 3][t]), z, 3)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 2):
seq(a(n), n=9..40);

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Series[
      b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]]+If[t==10, z, 1]*
      b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]], {z, 0, 3}]]];
a[n_] := Coefficient[b[2n, 0, 1], z, 2];
a /@ Range[9, 40] (* Jean-François Alcover, Dec 27 2020, after Alois P. Heinz *)

a(n) ~ c * d^n * sqrt(n), where d = 3.992152919721564592666177480042427843835641823811... is the root of equation 1 - 2*d + d^2 - 6*d^5 + 2*d^6 - 4*d^9 + d^10 = 0, and c = 0.00000109315704269290466088403991068... . - Vaclav Kotesovec, Jul 16 2014

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A243836 Number A(n,k) of Dyck paths of semilength n having exactly ten (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A243771 Number of Dyck paths of semilength n having exactly two (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1).

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A243872 Number of Dyck paths of semilength n having exactly 2 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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