cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 23 results. Next

A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (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); triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 11, 2, 9, 16, 12, 4, 1, 1, 57, 69, 5, 127, 161, 98, 35, 7, 1, 323, 927, 180, 1515, 1997, 1056, 280, 14, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 10455, 25638, 18357, 4115, 220, 1, 20705, 68850, 77685, 34840, 5685, 246, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 09 2014

Keywords

Examples

			Triangle T(n,k) begins:
: n\k :    0     1     2     3    4    5  ...
+-----+----------------------------------------------------------
:  0  :    1;                                 [row  0 of A131427]
:  1  :    0,    1;                           [row  1 of A131427]
:  2  :    0,    1,    1;                     [row  2 of A090181]
:  3  :    1,    3,    1;                     [row  3 of A001263]
:  4  :    1,   11,    2;                     [row  4 of A091156]
:  5  :    9,   16,   12,    4,   1;          [row  5 of A091869]
:  6  :    1,   57,   69,    5;               [row  6 of A091156]
:  7  :  127,  161,   98,   35,   7,   1;     [row  7 of A092107]
:  8  :  323,  927,  180;                     [row  8 of A091958]
:  9  : 1515, 1997, 1056,  280,  14;          [row  9 of A135306]
: 10  : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]
		

Crossrefs

A243827 Number A(n,k) of Dyck paths of semilength n having exactly one occurrence 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.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 4, 6, 1, 0, 0, 0, 0, 1, 2, 11, 10, 1, 0, 0, 0, 0, 0, 4, 6, 26, 15, 1, 0, 0, 0, 0, 0, 1, 11, 16, 57, 21, 1, 0, 0, 0, 0, 0, 1, 4, 26, 45, 120, 28, 1, 0, 0, 0, 0, 1, 1, 5, 15, 57, 126, 247, 36, 1, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			Square array A(n,k) begins:
  0, 0, 0,  0,   0,    0,   0,    0,    0,    0, ...
  1, 1, 1,  0,   0,    0,   0,    0,    0,    0, ...
  0, 0, 1,  1,   1,    1,   1,    0,    0,    0, ...
  0, 0, 1,  3,   4,    2,   4,    1,    1,    1, ...
  0, 0, 1,  6,  11,    6,  11,    4,    5,    5, ...
  0, 0, 1, 10,  26,   16,  26,   15,   21,   17, ...
  0, 0, 1, 15,  57,   45,  57,   50,   78,   54, ...
  0, 0, 1, 21, 120,  126, 120,  161,  274,  177, ...
  0, 0, 1, 28, 247,  357, 247,  504,  927,  594, ...
  0, 0, 1, 36, 502, 1016, 502, 1554, 3061, 1997, ...
		

Crossrefs

Columns k=2-10 give: A000012(n) for n>0, A000217(n-1) for n>0, A000295(n-1) for n>0, A005717(n-1) for n>1, A000295(n-1) for n>0, A014532(n-2) for n>2, A108863, A244235, A244236.
Main diagonal gives A243770 or column k=1 of A243752.

A243828 Number A(n,k) of Dyck paths of semilength n having exactly two (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 6, 10, 0, 0, 0, 0, 0, 1, 2, 20, 15, 0, 0, 0, 0, 0, 0, 3, 15, 50, 21, 0, 0, 0, 0, 0, 0, 2, 12, 69, 105, 28, 0, 0, 0, 0, 0, 0, 1, 15, 40, 252, 196, 36, 0, 0, 0, 0, 0, 0, 0, 5, 69, 135, 804, 336, 45, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  2, 2,  1,   0,    0,    0,    0,    0,   0,    0, ...
  0, 0,  3,   1,    0,    1,    0,    0,   0,    0, ...
  0, 0,  6,   6,    2,    3,    2,    1,   0,    0, ...
  0, 0, 10,  20,   15,   12,   15,    5,   0,    2, ...
  0, 0, 15,  50,   69,   40,   69,   24,   3,   15, ...
  0, 0, 21, 105,  252,  135,  252,   98,  28,   69, ...
  0, 0, 28, 196,  804,  441,  804,  378, 180,  273, ...
  0, 0, 36, 336, 2349, 1428, 2349, 1386, 954, 1056, ...
		

Crossrefs

Main diagonal gives A243771 or column k=2 of A243752.

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.

Original entry on oeis.org

1, 1, 2, 5, 13, 1, 37, 5, 112, 19, 1, 352, 70, 7, 1136, 259, 34, 1, 3742, 962, 149, 9, 12529, 3585, 627, 54, 1, 42513, 13399, 2584, 279, 11, 145868, 50201, 10529, 1334, 79, 1, 505234, 188481, 42606, 6092, 474, 13, 1764157, 709001, 171563, 27048, 2561, 109, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 03 2014

Keywords

Comments

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

Examples

			T(4,1) = 1: UDUUDUDD.
T(5,1) = 5: UDUDUUDUDD, UDUUDUDDUD, UDUUDUDUDD, UDUUDUUDDD, UUDUUDUDDD.
T(6,1) = 19: UDUDUDUUDUDD, UDUDUUDUDDUD, UDUDUUDUDUDD, UDUDUUDUUDDD, UDUUDUDDUDUD, UDUUDUDDUUDD, UDUUDUDUDDUD, UDUUDUDUDUDD, UDUUDUDUUDDD, UDUUDUUDDDUD, UDUUDUUDDUDD, UDUUDUUUDDDD, UUDDUDUUDUDD, UUDUDUUDUDDD, UUDUUDUDDDUD, UUDUUDUDDUDD, UUDUUDUDUDDD, UUDUUDUUDDDD, UUUDUUDUDDDD.
T(6,2) = 1: UDUUDUUDUDDD.
T(7,2) = 7: UDUDUUDUUDUDDD, UDUUDUDUUDUDDD, UDUUDUUDUDDDUD, UDUUDUUDUDDUDD, UDUUDUUDUDUDDD, UDUUDUUDUUDDDD, UUDUUDUUDUDDDD.
T(8,3) = 1: UDUUDUUDUUDUDDDD.
Triangle T(n,k) begins:
:  0 :     1;
:  1 :     1;
:  2 :     2;
:  3 :     5;
:  4 :    13,    1;
:  5 :    37,    5;
:  6 :   112,   19,   1;
:  7 :   352,   70,   7;
:  8 :  1136,  259,  34,  1;
:  9 :  3742,  962, 149,  9;
: 10 : 12529, 3585, 627, 54, 1;
		

Crossrefs

Row sums give A000108.
T(n,floor(n/2)-1) gives A093178(n) for n>3.
T(45,k) = A243752(45,k).
T(n,0) = A243753(n,45).

Programs

  • Maple
    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, 5, 2, 4][t])*
         `if`(t=6, z, 1) +b(x-1, y-1, [1, 3, 1, 3, 6, 1][t]))))
        end:
    T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
    seq(T(n), n=0..20);
  • Mathematica
    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 *)

A243829 Number A(n,k) of Dyck paths of semilength n having exactly three (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 20, 0, 0, 0, 0, 0, 0, 0, 10, 50, 0, 0, 0, 0, 0, 0, 1, 0, 50, 105, 0, 0, 0, 0, 0, 0, 0, 4, 5, 175, 196, 0, 0, 0, 0, 0, 0, 0, 0, 20, 56, 490, 336, 0, 0, 0, 0, 0, 0, 0, 1, 5, 80, 364, 1176, 540, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  5, 5,   1,    0,    0,    0,    0,   0,  0,   0, ...
  0, 0,   6,    1,    0,    1,    0,   0,  0,   0, ...
  0, 0,  20,   10,    0,    4,    0,   1,  0,   0, ...
  0, 0,  50,   50,    5,   20,    5,   6,  0,   0, ...
  0, 0, 105,  175,   56,   80,   56,  35,  0,   5, ...
  0, 0, 196,  490,  364,  315,  364, 168,  0,  49, ...
  0, 0, 336, 1176, 1800, 1176, 1800, 750, 12, 280, ...
		

Crossrefs

Main diagonal gives A243772 or column k=3 of A243752.

A243830 Number A(n,k) of Dyck paths of semilength n having exactly four (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 1, 50, 0, 0, 0, 0, 0, 0, 0, 0, 15, 175, 0, 0, 0, 0, 0, 0, 0, 1, 0, 105, 490, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 490, 1176, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 14, 1764, 2520, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 140, 210, 5292, 4950, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  14, 14,    1,    0,   0,   0,   0,   0, 0,  0, ...
   0,  0,   10,    1,   0,   1,   0,   0, 0,  0, ...
   0,  0,   50,   15,   0,   5,   0,   1, 0,  0, ...
   0,  0,  175,  105,   0,  30,   0,   7, 0,  0, ...
   0,  0,  490,  490,  14, 140,  14,  48, 0,  0, ...
   0,  0, 1176, 1764, 210, 630, 210, 264, 0, 14, ...
		

Crossrefs

Main diagonal gives A243773 or column k=4 of A243752.

A243831 Number A(n,k) of Dyck paths of semilength n having exactly five (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 1, 105, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 490, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 196, 1764, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 1176, 5292, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 5292, 13860, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  42, 42,    1,    0,  0,    0,  0,   0, 0, 0,   0, ...
   0,  0,   15,    1,  0,    1,  0,   0, 0, 0,   1, ...
   0,  0,  105,   21,  0,    6,  0,   1, 0, 0,   1, ...
   0,  0,  490,  196,  0,   42,  0,   8, 0, 0,  13, ...
   0,  0, 1764, 1176,  0,  224,  0,  63, 0, 0,  52, ...
   0,  0, 5292, 5292, 42, 1134, 42, 390, 0, 0, 244, ...
		

Crossrefs

Main diagonal gives A243774 or column k=5 of A243752.

A243832 Number A(n,k) of Dyck paths of semilength n having exactly six (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 0, 0, 0, 0, 0, 0, 132, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 196, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 1176, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 336, 5292, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 2520, 19404, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  132, 132,    1,    0, 0,   0, 0,  0, 0, 0,  0, ...
    0,   0,   21,    1, 0,   1, 0,  0, 0, 0,  1, ...
    0,   0,  196,   28, 0,   7, 0,  1, 0, 0,  1, ...
    0,   0, 1176,  336, 0,  56, 0,  9, 0, 0, 15, ...
    0,   0, 5292, 2520, 0, 336, 0, 80, 0, 0, 64, ...
		

Crossrefs

Main diagonal gives A243775 or column k=6 of A243752.

A243833 Number A(n,k) of Dyck paths of semilength n having exactly seven (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 429, 0, 0, 0, 0, 0, 0, 0, 429, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 336, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 2520, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 540, 13860, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  429, 429,    1,   0, 0,  0, 0,  0, 0, 0,  0, ...
    0,   0,   28,   1, 0,  1, 0,  0, 0, 0,  1, ...
    0,   0,  336,  36, 0,  8, 0,  1, 0, 0,  1, ...
    0,   0, 2520, 540, 0, 72, 0, 10, 0, 0, 17, ...
		

Crossrefs

Main diagonal gives A243776 or column k=7 of A243752.

A243834 Number A(n,k) of Dyck paths of semilength n having exactly eight (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.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1430, 0, 0, 0, 0, 0, 0, 0, 0, 1430, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 540, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 45, 4950, 0, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Examples

			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, ...
  1430, 1430,    1,   0, 0,  0, 0,  0, 0, 0,  0, 0, ...
     0,    0,   36,   1, 0,  1, 0,  0, 0, 0,  1, 0, ...
     0,    0,  540,  45, 0,  9, 0,  1, 0, 0,  1, 0, ...
     0,    0, 4950, 825, 0, 90, 0, 11, 0, 0, 19, 0, ...
		

Crossrefs

Main diagonal gives A243777 or column k=8 of A243752.
Showing 1-10 of 23 results. Next