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.

Previous Showing 31-40 of 41 results. Next

A243772 Number of Dyck paths of semilength n having exactly three (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

4, 5, 35, 0, 280, 1991, 4115, 34840, 96286, 309036, 1045200, 193240, 5159120, 40653929, 105545340, 603157520, 2582073261, 11015773404, 26828044860, 182118031760, 726122370210, 3026319516720, 9620891607824, 49247195403600, 161316665871200, 638742288482240
Offset: 5

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(5) = 4: UDUDUDUUDD, UDUDUUDUDD, UDUUDUDUDD, UUDUDUDUDD.
a(6) = 5: UUDDUUDDUUDD, UUDDUUDUUDDD, UUDUUDDDUUDD, UUDUUDDUUDDD, UUDUUDUUDDDD.

Links

Crossrefs

Column k=3 of A243752.
Main diagonal of A243829.

A243773 Number of Dyck paths of semilength n having exactly four (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, 0, 7, 0, 14, 781, 220, 5685, 12104, 10920, 471900, 140, 402920, 10257990, 17056745, 82403580, 1152019163, 3595341849, 2413091295, 40392418220, 235442534915, 804119873760, 2973203848860, 15274217567340, 32286978092250, 94142466440960, 909684828893676
Offset: 5

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(5) = 1: UDUDUDUDUD.
a(7) = 7: UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD.

Links

Crossrefs

Column k=4 of A243752.
Main diagonal of A243830.

A243774 Number of Dyck paths of semilength n having exactly five (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, 0, 0, 244, 1, 246, 379, 0, 179790, 0, 5096, 1721037, 1595288, 3992352, 455454822, 842632268, 63280332, 4243256192, 50749141182, 109671572304, 614565138768, 2580550363440, 2759439518892, 4369338844416, 474848703434955, 110506726968, 50798936389648
Offset: 7

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(7) = 1: UUUUUUUDDDDDDD.
a(11) = 1: UDUUDUUDUUDUUDUUDDDDDD.

Links

Crossrefs

Column k=5 of A243752.
Main diagonal of A243831.

A243775 Number of Dyck paths of semilength n having exactly six (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

64, 0, 1, 1, 0, 58310, 0, 0, 169352, 82494, 40040, 163083704, 140035157, 306726, 178134488, 7368664408, 6899287752, 86343111918, 221762412480, 89389210680, 45967495392, 229706136771072, 18828656, 987116153856, 88775675895264, 303516229481536, 10355664669739531
Offset: 10

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(12) = 1: UUDDUUDDUUDDUUDDUUDDUUDD.
a(13) = 1: UUDUUDUUDUUDUUDUUDUDDDDDDD.

Links

Crossrefs

Column k=6 of A243752.
Main diagonal of A243832.

A243776 Number of Dyck paths of semilength n having exactly seven (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

17, 0, 0, 0, 0, 16128, 0, 0, 6695, 2100, 0, 53635915, 15338708, 72, 2004080, 715058696, 158698944, 8215899828, 8360997120, 861521760, 42073200, 103648453795647, 0, 4069967184, 3363112380658, 16806651486228, 1514241262404859, 1140768813493596, 18600198213831560
Offset: 10

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(10) = 17: UDUDUDUDUDUDUDUDUUDD, UDUDUDUDUDUDUDUUDUDD, UDUDUDUDUDUDUUDUDUDD, UDUDUDUDUDUUDUDUDUDD, UDUDUDUDUUDUDUDUDUDD, UDUDUDUUDUDUDUDUDUDD, UDUDUUDUDUDUDUDUDUDD, UDUUDUDUDUDUDUDUDUDD, UUDDUDUDUDUDUDUDUDUD, UUDUDDUDUDUDUDUDUDUD, UUDUDUDDUDUDUDUDUDUD, UUDUDUDUDDUDUDUDUDUD, UUDUDUDUDUDDUDUDUDUD, UUDUDUDUDUDUDDUDUDUD, UUDUDUDUDUDUDUDDUDUD, UUDUDUDUDUDUDUDUDDUD, UUUDUDUDUDUDUDUDUDDD.

Links

Crossrefs

Column k=7 of A243752.
Main diagonal of A243833.

A243777 Number of Dyck paths of semilength n having exactly eight (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, 0, 0, 0, 0, 3780, 0, 0, 46, 20, 0, 16356705, 910111, 0, 1430, 45077670, 755820, 518221227, 101454300, 1479060, 0, 43816171893552, 0, 420732, 54663267285, 583180547204, 181720848039484, 60770152697265, 2020797438130380, 52163677215449, 31943272670595
Offset: 10

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(10) = 1: UUDUDUDUDUDUDUDUDUDD.

Links

Crossrefs

Column k=8 of A243752.
Main diagonal of A243834.

A243778 Number of Dyck paths of semilength n having exactly nine (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, 0, 0, 0, 0, 740, 0, 0, 0, 0, 0, 4655685, 19546, 0, 0, 1747955, 0, 20759960, 167960, 110, 0, 17405076899616, 0, 0, 261107026, 11058883809, 17817753939932, 2065690806135, 180987639660985, 533693067696, 45880541850, 2285677991937600, 16275024211846481596
Offset: 10

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Examples

			a(10) = 1: UDUDUDUDUDUDUDUDUDUD.

Links

Crossrefs

Column k=9 of A243752.
Main diagonal of A243835.

A243779 Number of Dyck paths of semilength n having exactly ten (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

120, 0, 0, 0, 0, 0, 1242286, 67, 0, 0, 37818, 0, 490210, 0, 0, 0, 6509142657552, 0, 0, 163318, 81043930, 1409799253711, 38901493532, 13399508027211, 2055406724, 1430715, 75368838054510, 5436661972988347050, 58576501701861, 1142005789397748, 7372114735568176431
Offset: 15

Views

Author

Alois P. Heinz, Jun 10 2014

Keywords

Links

Crossrefs

Column k=10 of A243752.
Main diagonal of A243836.

A135305 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UUUU's.

Original entry on oeis.org

1, 1, 2, 5, 13, 1, 36, 5, 1, 104, 21, 6, 1, 309, 84, 28, 7, 1, 939, 322, 124, 36, 8, 1, 2905, 1206, 522, 174, 45, 9, 1, 9118, 4455, 2127, 795, 235, 55, 10, 1, 28964, 16302, 8492, 3487, 1155, 308, 66, 11, 1, 92940, 59268, 33396, 14894, 5412, 1617, 394, 78, 12, 1
Offset: 0

Views

Author

N. J. A. Sloane, Dec 07 2007

Keywords

Comments

Each of rows 0, 1, 2, 3 has one entry. Row n (n >= 3) has n-2 entries. Row sums yield the Catalan numbers (A000108). Column 0 yields A036765. - Emeric Deutsch, Dec 14 2007

Examples

			Triangle begins:
1
1
2
5
13 1
36 5 1
104 21 6 1
309 84 28 7 1
...
T(5,1) = 5 because we have UUUUDUDDDD, UUUUDDUDDD, UUUUDDDUDD, UUUUDDDDUD and UDUUUUDDDD.

Links

Crossrefs

Cf. A000108, A036765, A243752.

Programs

  • Maple
    eq:=(1-t)*z^3*G^3+z*(t+z-t*z)*G^2+((1-t)*z-1)*G+1: g:=RootOf(eq,G): gser:= simplify(series(g,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(gser,z,n)) end do: 1;1;2; for n from 3 to 12 do seq(coeff(P[n],t,j),j=0..n-3) end do; # yields sequence in triangular form; # Emeric Deutsch, Dec 14 2007
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y+1, min(t+1, 4))*
      `if`(t=4, z, 1) +b(x-1, y-1, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 02 2014
  • 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, Min[t+1, 4]]*If[t == 4, z, 1] + b[x-1, y-1, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]] @ b[2*n, 0, 1]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Nov 28 2014, after Alois P. Heinz *)

Formula

G.f.: G=G(t,z) satisfies (1-t)*z^3*G^3 + z*(t+z-t*z)*G^2 + ((1-t)*z-1)*G+1 = 0. - Emeric Deutsch, Dec 14 2007

Extensions

Edited and extended by Emeric Deutsch, Dec 14 2007

A243838 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDDUUUUDUDDDDUDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/9)), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 1, 58783, 3, 208002, 10, 742865, 35, 2674314, 126, 9694383, 462, 35355954, 1716, 129638355, 6435, 477614390, 24310, 1767170813, 92376, 1, 6563767708, 352708, 4, 24464914958, 1352046, 16, 91477363405, 5200170, 65
Offset: 0

Views

Author

Alois P. Heinz, Jun 11 2014

Keywords

Comments

UDUUDDUUUUDUDDDDUDUD is a Dyck path that contains all sixteen consecutive step patterns of length 4.

Examples

			Triangle T(n,k) begins:
:  0 :           1;
:  1 :           1;
:  2 :           2;
:  3 :           5;
:  4 :          14;
:  5 :          42;
:  6 :         132;
:  7 :         429;
:  8 :        1430;
:  9 :        4862;
: 10 :       16795,       1;
: 11 :       58783,       3;
: 12 :      208002,      10;
: 13 :      742865,      35;
: 14 :     2674314,     126;
: 15 :     9694383,     462;
: 16 :    35355954,    1716;
: 17 :   129638355,    6435;
: 18 :   477614390,   24310;
: 19 :  1767170813,   92376,  1;
: 20 :  6563767708,  352708,  4;
: 21 : 24464914958, 1352046, 16;

Links

Crossrefs

Row sums give A000108.
T(736522,k) = A243752(736522,k).
T(n,0) = A243753(n,736522).
Cf. A243820.

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,
       8, 9, 10, 11, 2, 13, 5, 4, 2, 2, 18, 2, 20, 5][t])
      +`if`(t=20, z, 1) *b(x-1, y-1, [1, 3, 1, 3, 6, 7,
       1, 3, 3, 3, 12, 1, 14, 15, 16, 17, 1, 19, 1, 3][t]))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..30);
  • Mathematica
    b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Expand[If[y >= x - 1, 0, b[x - 1, y + 1, {2, 2, 4, 5, 2, 4, 8, 9, 10, 11, 2, 13, 5, 4, 2, 2, 18, 2, 20, 5}[[t]]]] + If[t == 20, z, 1]*If[y == 0, 0, b[x - 1, y - 1, {1, 3, 1, 3, 6, 7, 1, 3, 3, 3, 12, 1, 14, 15, 16, 17, 1, 19, 1, 3}[[t]]]]]];
T[n_] := CoefficientList[b[2n, 0, 1], z];
T /@ Range[0, 30] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)
Previous Showing 31-40 of 41 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.