A243881 -id:A243881 - cp's OEIS frontend

A243870 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

1, 1, 2, 5, 14, 41, 129, 419, 1395, 4737, 16338, 57086, 201642, 718855, 2583149, 9346594, 34023934, 124519805, 457889432, 1690971387, 6268769864, 23320702586, 87031840257, 325741788736, 1222429311437, 4598725914380, 17339388194985, 65514945338284

Offset: 0

Alois P. Heinz, Jun 13 2014

nonn

UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.

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

Column k=0 of A243881.

Column k=738 of A243753.

a:= proc(n) option remember; `if`(n<14, [1, 1, 2, 5, 14, 41,
       129, 419, 1395, 4737, 16338, 57086, 201642, 718855][n+1],
       ((4*n-2)*a(n-1) -(3*n-9)*a(n-4) +(10*n-41)*a(n-5)
       -(3*n-21)*a(n-8) +(8*n-64)*a(n-9) -(n-14)*a(n-10)
       -(n-11)*a(n-12) +(2*n-25)*a(n-13) +(14-n)*a(n-14))/(n+1))
    end:
seq(a(n), n=0..40);

a[n_] := a[n] = If[n<14, {1, 1, 2, 5, 14, 41, 129, 419, 1395, 4737, 16338, 57086, 201642, 718855}[[n+1]], ((4n-2)a[n-1] - (3n-9)a[n-4] + (10n-41)a[n-5] - (3n-21)a[n-8] + (8n-64)a[n-9] - (n-14)a[n-10] - (n-11)a[n-12] + (2n-25)a[n-13] + (14-n)a[n-14])/(n+1)];
a /@ Range[0, 40] (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Recursion: see Maple program.

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)).

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

A243873 Number of Dyck paths of semilength n having exactly 3 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 5, 23, 105, 472, 2118, 9446, 41847, 184256, 806740, 3514298, 15238732, 65803650, 283077978, 1213561196, 5186141801, 22098720181, 93913940321, 398127653185, 1683928072645, 7107304159469, 29938529102885, 125880340885997, 528371537192555, 2214227613719264

Offset: 13

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 13..400

Column k=3 of A243881.

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, 4)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 3):
seq(a(n), n=13..45);

A243874 Number of Dyck paths of semilength n having exactly 4 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 6, 31, 156, 766, 3717, 17812, 84342, 395152, 1833853, 8438976, 38540936, 174819086, 788082431, 3532770025, 15755543925, 69937932805, 309113716505, 1360804143915, 5968626187120, 26089764842864, 113680654898844, 493874661384094, 2139660006480909

Offset: 17

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 17..400

Column k=4 of A243881.

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, 5)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 4):
seq(a(n), n=17..45);

A243875 Number of Dyck paths of semilength n having exactly 5 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 7, 40, 219, 1161, 6035, 30816, 154815, 766711, 3749225, 18128129, 86772929, 411599785, 1936434085, 9042584447, 41939926492, 193310490160, 885917766448, 4038628790596, 18320941855600, 82734637234636, 372039593944604, 1666387342165538, 7436328773819975

Offset: 21

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 21..400

Column k=5 of A243881.

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, 6)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 5):
seq(a(n), n=21..50);

A243876 Number of Dyck paths of semilength n having exactly 6 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 8, 50, 295, 1674, 9255, 50037, 265190, 1381151, 7083239, 35832547, 179064335, 885033494, 4330974280, 21002926804, 101014451257, 482163988802, 2285470580378, 10763603536650, 50390267987583, 234599001141494, 1086577533281204, 5008393400154248

Offset: 25

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 25..400

Column k=6 of A243881.

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, 7)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 6):
seq(a(n), n=25..55);

A243877 Number of Dyck paths of semilength n having exactly 7 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 9, 61, 385, 2323, 13583, 77363, 430573, 2348528, 12584052, 66372328, 345160962, 1772302098, 8996192858, 45189272314, 224832198163, 1108842335240, 5424622033040, 26340438937256, 127018289627132, 608569050945950, 2898295732654434, 13725710735084610

Offset: 29

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 29..400

Column k=7 of A243881.

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, 8)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 7):
seq(a(n), n=29..60);

A243878 Number of Dyck paths of semilength n having exactly 8 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 10, 73, 490, 3127, 19249, 115021, 669745, 3812716, 21277658, 116666435, 629665898, 3350420024, 17599292330, 91368992279, 469293511892, 2386777084592, 12029136326922, 60118399193577, 298121360285805, 1467661404628893, 7176555449003580, 34870090954789419

Offset: 33

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 33..400

Column k=8 of A243881.

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, 9)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 8):
seq(a(n), n=33..60);

A243879 Number of Dyck paths of semilength n having exactly 9 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

Original entry on oeis.org

1, 11, 86, 611, 4106, 26508, 165608, 1005693, 5958138, 34538560, 196383607, 1097479232, 6038391492, 32757730552, 175436127352, 928559489820, 4861821384020, 25202877769350, 129444778524955, 659155791410730, 3329785315219783, 16695460286688023, 83126852562101708

Offset: 37

Alois P. Heinz, Jun 13 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 37..400

Column k=9 of A243881.

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, 10)))
    end:
a:= n-> coeff(b(2*n, 0, 1), z, 9):
seq(a(n), n=37..65);

cp's OEIS Frontend

A243870 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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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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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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A243873 Number of Dyck paths of semilength n having exactly 3 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A243874 Number of Dyck paths of semilength n having exactly 4 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A243875 Number of Dyck paths of semilength n having exactly 5 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A243876 Number of Dyck paths of semilength n having exactly 6 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A243877 Number of Dyck paths of semilength n having exactly 7 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A243878 Number of Dyck paths of semilength n having exactly 8 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).

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A243879 Number of Dyck paths of semilength n having exactly 9 (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)).