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.

User: Sarah Selkirk

Sarah Selkirk's wiki page.

Sarah Selkirk has authored 8 sequences.

A334980 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_2-Dyck paths of length 4*n. A 3_2-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

Original entry on oeis.org

0, 3, 31, 248, 1941, 15334, 122915, 999456, 8231740, 68562887, 576661761, 4891506968, 41801697070, 359574305580, 3111012673755, 27055673506128, 236387476114548, 2073957836402524, 18264689865840284, 161403223665821280, 1430768729986730685, 12719497076318052990
Offset: 0

Views

Author

Sarah Selkirk, May 18 2020

Keywords

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 3 is the total number of down steps before the first up step.

Examples

			For n = 2, the 3_2-Dyck paths are UDDDDDUD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDDUD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, DDUDDDUD, DDUDDUDD, DDUDUDDD, DDUUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 5 + 4 + 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 31.

Links

Crossrefs

Cf. A334976, A334977, A334978, A334979.

Programs

  • Mathematica
    a[0] = 0; a[n_] := 3*Binomial[4*n+7, n+1]/(4*n + 7) - 12 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 22, 0]
  • SageMath
    [3*binomial(4*n + 7, n + 1)/(4*n + 7) - 12*binomial(4*n + 3, n)/(4*n + 3) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Formula

a(0) = 0 and a(n) = 3*binomial(4*n+7, n+1)/(4*n+7) - 12*binomial(4*n+3, n)/(4*n+3) for n > 0.

A334979 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

Original entry on oeis.org

0, 1, 16, 132, 1034, 8134, 64880, 525132, 4307512, 35750473, 299759200, 2535849836, 21619615164, 185582339740, 1602675301920, 13915031036412, 121396437548136, 1063653520870612, 9355905795325888, 82585983533819920, 731350409249262330, 6495673923406863630
Offset: 0

Views

Author

Sarah Selkirk, May 18 2020

Keywords

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 1 is the total number of down steps before the first up step.

Examples

			For n = 2, the 3_1-Dyck paths are UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 16.

Links

Crossrefs

Cf. A334976, A334977, A334978, A334980.

Programs

  • Mathematica
    a[0] = 0; a[n_] := Binomial[4*n+6, n+1]/(2*n + 3) - 4 * Binomial[4*n + 2, n]/(2*n + 1); Array[a, 22, 0]
  • SageMath
    [binomial(4*n + 6, n + 1)/(2*n + 3) - 4*binomial(4*n + 2, n)/(2*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Formula

a(0) = 0 and a(n) = binomial(4*n+6, n+1)/(2*n+3) - 4*binomial(4*n+2, n)/(2*n+1) for n > 0.
G.f.: ((1 - 4*x)*hypergeom([1/2, 3/4, 5/4], [4/3, 5/3], 2^8*x/3^3) - 1 + 2*x)/x. - Stefano Spezia, Aug 25 2025

A334978 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3-Dyck paths of length 4*n. A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

Original entry on oeis.org

0, 0, 6, 52, 409, 3208, 25484, 205452, 1679332, 13894848, 116193246, 980658172, 8343605534, 71492410640, 616418176920, 5344364518140, 46565472754044, 407529832131712, 3580911446989368, 31579384975219920, 279414033129153065, 2479725948121016040
Offset: 0

Views

Author

Sarah Selkirk, May 18 2020

Keywords

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 0 is the total number of down steps before the first up step.

Examples

			For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. Therefore, the total number of down steps between the first and second up steps is a(2) = 3 + 2 + 1 + 0 = 6.

Links

Crossrefs

Cf. A334976, A334977, A334979, A334980.

Programs

  • Mathematica
    a[0] = 0; a[n_] := Binomial[4*n+5, n+1]/(4*n + 5) - 4 * Binomial[4*n + 1, n]/(4*n + 1); Array[a, 22, 0]
  • SageMath
    [binomial(4*n + 5, n + 1)/(4*n + 5) - 4*binomial(4*n + 1, n)/(4*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Formula

a(0) = 0 and a(n) = binomial(4*n+5, n+1)/(4*n+5) - 4*binomial(4*n+1, n)/(4*n+1) for n > 0.

A334977 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

Original entry on oeis.org

0, 1, 9, 53, 299, 1692, 9690, 56221, 330165, 1959945, 11745435, 70974252, 432019844, 2646716264, 16307880462, 100996570221, 628356589721, 3925544432355, 24616047166095, 154886752443885, 977595783524955, 6187863825170160, 39269844955755960, 249819662230403148
Offset: 0

Views

Author

Sarah Selkirk, May 18 2020

Keywords

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 1 is the total number of down steps before the first up step.

Examples

			For n = 2, the 2_1-Dyck paths are UDDDUD, UDDUDD, UDUDDD, UUDDDD, DUDDUD, DUDUDD, DUUDDD. Therefore the total number of down steps between the first and second up step is a(2) = 3 + 2 + 1 + 0 + 2 + 1 +0 = 9.

Links

Crossrefs

Cf. A334976, A334978, A334979, A334980.

Programs

  • Mathematica
    a[0] = 0; a[n_] := 2*Binomial[3*n+5, n+1]/(3*n + 5) - 6 * Binomial[3*n + 2, n]/(3*n + 2); Array[a, 24, 0]
  • SageMath
    [2*binomial(3*n + 5, n + 1)/(3*n + 5) - 6*binomial(3*n + 2, n)/(3*n + 2) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Formula

a(0) = 0 and a(n) = 2*binomial(3*n+5, n+1)/(3*n+5) - 6*binomial(3*n+2, n)/(3*n+2) for n > 0.

A334976 a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative path with steps (1, 2), (1, -1) that starts and ends at y = 0.

Original entry on oeis.org

0, 0, 3, 19, 108, 609, 3468, 20007, 116886, 690690, 4122495, 24823188, 150629248, 920274804, 5656456104, 34954487967, 217044280458, 1353539406660, 8474029162305, 53241343026795, 335592121524660, 2121577490385885, 13448859209014320, 85467026778421860
Offset: 0

Views

Author

Sarah Selkirk, May 18 2020

Keywords

Comments

For n = 1, there is no (n-1)-th up step, a(1) = 0 is the total number of down steps before the first up step.

Examples

			For n = 2, the 2-Dyck paths are UDDUDD, UDUDDD, UUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 2+1+0 = 3.

Links

Crossrefs

Cf. A334977, A334978, A334979, A334980.

Programs

  • Maple
    alias(PS=ListTools:-PartialSums): A334976List := proc(m) local A, P, n;
A := [0,0]; P := [1,0]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A334976List(24); # Peter Luschny, Mar 26 2022
  • Mathematica
    a[0] = 0; a[n_] := Binomial[3*n+4, n+1]/(3*n + 4) - 3 * Binomial[3*n + 1, n]/(3*n + 1); Array[a, 24, 0]
  • SageMath
    [binomial(3*n + 4, n + 1)/(3*n + 4) - 3*binomial(3*n + 1, n)/(3*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020

Formula

a(0) = 0 and a(n) = binomial(3*n+4, n+1)/(3*n+4) - 3*binomial(3*n+1, n)/(3*n+1) for n > 0.

A334787 a(n) is the total number of down steps before the first up step in all 4_3-Dyck paths of length 5*n. A 4_3-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -3.

Original entry on oeis.org

0, 6, 34, 251, 2105, 19040, 181076, 1784728, 18067803, 186754590, 1962728460, 20910164730, 225308533359, 2451112021568, 26885549373440, 297008527319440, 3301615350645935, 36903975448964670, 414518195957729886, 4676429192392769805, 52965796433899543810
Offset: 0

Views

Author

Sarah Selkirk, May 11 2020

Keywords

Examples

			For n = 1, there are the 4_3-Dyck paths UDDDD, DUDDD, DDUDD, DDDUD. Before the first up step there are a(1) = 0 + 1 + 2 + 3 = 6 down steps in total.

Links

Crossrefs

Cf. A001764, A002293, A002294, A334785, A334786.

Programs

  • Mathematica
    a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n+1) - Binomial[5*n+3, n]/(n+1); Array[a, 21, 0]

Formula

a(0) = 0 and a(n) = 4*binomial(5*n, n)/(n+1) - binomial(5*n+3, n)/(n+1) for n > 0.
a(n) ~ c*2^(-8*n)*5^(5*n)/n^(3/2), where c = (131/128)*sqrt(5/(2*Pi)). - Stefano Spezia, Oct 19 2022

A334786 a(n) is the total number of down steps before the first up step in all 4_2-Dyck paths of length 5*n. A 4_2-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

Original entry on oeis.org

0, 3, 16, 115, 950, 8510, 80388, 788392, 7950930, 81935425, 859005840, 9132977490, 98240702586, 1067197649840, 11691092372000, 129011823098160, 1432744619523530, 16000911127589355, 179590878292003200, 2024687100104286525, 22917687021180660940
Offset: 0

Views

Author

Sarah Selkirk, May 11 2020

Keywords

Examples

			For n = 1, there are the 4_2-Dyck paths UDDDD, DUDDD, DDUDD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total.

Links

Crossrefs

Cf. A001764, A002293, A002294, A334785, A334787.

Programs

  • Mathematica
    a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n+1) - 2 * Binomial[5*n+2, n]/(n+1); Array[a, 21, 0]

Formula

a(0) = 0 and a(n) = 4 * binomial(5*n, n)/(n+1) - 2 * binomial(5*n+2, n)/(n+1) for n > 0.
a(n) ~ c*2^(-8*n)*5^(5*n)/n^(3/2), where c = (7/16)*sqrt(5/(2*Pi)). - Stefano Spezia, Oct 19 2022

A334785 a(n) is the total number of down steps before the first up step in all 3_2-Dyck paths of length 4*n. A 3_2-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -2.

Original entry on oeis.org

0, 3, 13, 74, 480, 3363, 24794, 189540, 1488744, 11941820, 97412601, 805602850, 6738919408, 56918898330, 484750343700, 4158094853640, 35891774969112, 311529010178628, 2717299393716836, 23806014817182600, 209389427777770240, 1848322153489496355
Offset: 0

Views

Author

Sarah Selkirk, May 11 2020

Keywords

Examples

			For n = 1, there are the 3_2-Dyck paths UDDD, DUDD, DDUD. Before the first up step there are a(1) = 0 + 1 + 2 = 3 down steps in total.

Links

Crossrefs

Cf. A001764, A002293, A002294, A334786, A334787.

Programs

  • Mathematica
    a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n+1) - Binomial[4*n+2, n]/(n+1); Array[a, 22, 0]

Formula

a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) for n > 0.
a(n) ~ c*2^(8*n)*3^(-3*n)/n^(3/2), where c = (11/9)*sqrt(2/(3*Pi)). - Stefano Spezia, Oct 19 2022

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

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