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 1-4 of 4 results.

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

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.

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.

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.
Showing 1-4 of 4 results.

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