A243827 -id:A243827 - 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.

A243770 Number of Dyck paths of semilength n having exactly one occurrence 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, 3, 11, 16, 57, 161, 927, 1997, 5539, 25638, 68850, 275765, 995088, 2784600, 19235059, 53549250, 177389053, 711629836, 2641203240, 7517769634, 31706388438, 147201204924, 455738363552, 1614252170849, 6020919907344, 23811404216400, 79787485940824

Offset: 1

Alois P. Heinz, Jun 10 2014

nonn

			a(1) = 1: (U)D.
a(2) = 1: U(UD)D.
a(3) = 3: UD(UU)DD, (UU)DDUD, (UU)DUDD.
a(4) = 11: UDUDU(UDD), UDU(UDD)UD, UDUUD(UDD), UDUU(UDD)D, U(UDD)UDUD, UUD(UDD)UD, UUDUD(UDD), UUDU(UDD)D, UU(UDD)DUD, UUUD(UDD)D, UUU(UDD)DD.

Alois P. Heinz, Table of n, a(n) for n = 1..400

Column k=1 of A243752.

Main diagonal of A243827.

A244236 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDUD.

Original entry on oeis.org

0, 0, 1, 1, 5, 14, 46, 150, 495, 1651, 5539, 18692, 63356, 215556, 735717, 2517941, 8637881, 29693938, 102263818, 352762106, 1218634659, 4215351719, 14598518663, 50611799048, 175639493624, 610076726280, 2120837219465, 7378415912617, 25687819032237

Offset: 0

Alois P. Heinz, Jun 23 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=1 of A094507 and column k=10 of A243827.

a:= proc(n) option remember; `if`(n<5, [0$2, 1$2, 5][n+1],
     ((n-2)*(2*n-7)^2*a(n-1) +(28*n^3-212*n^2+501*n-361)*a(n-2)
      +(28*n^3-208*n^2+481*n-344)*a(n-3) +(n-3)*(2*n-3)^2*a(n-4)
      -(n-4)*(2*n-3)*(2*n-5)*a(n-5)) / ((n-1)*(2*n-5)*(2*n-7)))
    end:
seq(a(n), n=0..30);

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[2 n, 0, 1], z, 1];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz in A094507 *)

a(n) ~ c * (1/2+sqrt(2)+sqrt(5+4*sqrt(2))/2)^n / sqrt(n), where c = 0.0543819313385500572292392822783525275532509057751364636784836521... . - Vaclav Kotesovec, Jul 16 2014

A108863 Number of Dyck paths containing exactly one UUUD.

Original entry on oeis.org

0, 0, 0, 1, 5, 21, 78, 274, 927, 3061, 9933, 31824, 100972, 317942, 995088, 3099105, 9612735, 29715525, 91595391, 281643480, 864189486, 2646805668, 8093543439, 24713953515, 75370741506, 229604257846, 698754428388, 2124616182139

Offset: 0

David Callan, Jul 25 2005

nonn

a(n) = number of Dyck n-paths containing exactly one UUUD.

Conjecture: this is the Motzkin transform of the sequence of three zeros followed by A001651. - R. J. Mathar, Dec 11 2008

			a(4) = 5 because UUUUDDDD, UUUDUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD
each contain one UUUD.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. same as A055219 except for offset and is column k=1 of A091958. Dyck paths containing no UUUD are counted by the Motzkin numbers (A001006).

Column k=8 of A243827.

CoefficientList[Series[(x-1+(1-2*x)*(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))/(x*(1-3*x)*(1+x*(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))),{x,0,20}],x] (* Vaclav Kotesovec, Mar 22 2014 *)

G.f. (x-1+(1-2*x)M)/(x(1-3*x)(1+x*M)) = Sum_{n>=0}a(n)x^n where M = (1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2) is the gf for Motzkin numbers (A001006); satisfies z^3 = (1 + z)(1 - 3z)( (1 - 3z + z^2)G + z^2(1 - 3z)G^2 ).
Recurrence: (n-3)*(n+2)*a(n) = (n+1)*(5*n-14)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) - 9*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ 3^n/2 * (1-5*sqrt(3)/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Mar 22 2014

A243871 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 3, 10, 35, 124, 454, 1684, 6305, 23781, 90209, 343809, 1315499, 5050144, 19442366, 75034354, 290203076, 1124511549, 4364693311, 16966567970, 66041815437, 257378634365, 1004167036295, 3921726323436, 15330264382726, 59977821022143, 234839855088313

Offset: 5

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=1 of A243881.

Column k=738 of A243827.

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, 2)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 1):
seq(a(n), n=5..40);

a(n) = (2*(2*n-17) *(2*n-19) *(2*n-9) *a(n-1) -(2*n-19) *(6*n^2-75*n+208) *a(n-4) +2*(2*n-17) *(10*n^2-136*n+387) *a(n-5) -(2*n-19) *(6*n^2-75*n+212) *a(n-8) +(32*n^3-704*n^2+4940*n-10850) *a(n-9) -(2*n-17) *(2*n-9) *(n-14) *a(n-10) -(2*n-19) *(n-8) *(2*n-9) *a(n-12) +2*(2*n-9) *(2*n^2-36*n+161) *a(n-13) -(n-10) *(2*n-17) *(2*n-9) *a(n-14)) / ((2*n-17) *(2*n-19) *(n-4)).

A244235 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDDU.

Original entry on oeis.org

0, 0, 0, 1, 5, 17, 54, 177, 594, 1997, 6698, 22487, 75701, 255455, 863576, 2923806, 9913448, 33658109, 114417190, 389385699, 1326522885, 4523352061, 15437800028, 52730424194, 180244620903, 616546133055, 2110330086114, 7227665869122, 24768041790134

Offset: 0

Alois P. Heinz, Jun 23 2014

nonn

			a(3) = 1: UUDDUD.
a(4) = 5: UDUUDDUD, UUDDUDUD, UUDDUUDD, UUDUDDUD, UUUDDUDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=9 of A243827.

a:= proc(n) option remember; `if`(n<4, binomial(n, 3),
     (2*(n-1)*(112*n^5-1220*n^4+5251*n^3-11122*n^2+11566*n-4764)*a(n-1)
     +(n-2)*(560*n^5-5820*n^4+23159*n^3-44070*n^2+40253*n-14010)*a(n-2)
     -6*(n-2)*(n-3)*(112*n^4-884*n^3+2437*n^2-2436*n+486)*a(n-3)
     +23*(n-2)*(n-3)*(n-4)*(112*n^3-492*n^2+623*n-267)*a(n-4)) /
     (n*(n-1)*(n-3)*(112*n^3-828*n^2+1943*n-1494)))
    end:
seq(a(n), n=0..30);

a(n) ~ c * ((1+sqrt(13+16*sqrt(2)))/2)^n / sqrt(n), where c = 0.09016594515129336503624934471608236212385331150935643095582327... . - Vaclav Kotesovec, Jul 16 2014

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.

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A243770 Number of Dyck paths of semilength n having exactly one occurrence 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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A244236 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDUD.

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A108863 Number of Dyck paths containing exactly one UUUD.

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A243871 Number of Dyck paths of semilength n having exactly 1 occurrence of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A244235 Number of Dyck paths of semilength n having exactly one occurrence of the consecutive pattern UDDU.

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