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.

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

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

A334612 a(n) is the total number of down-steps after the final up-step in all 4_3-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

Original entry on oeis.org

0, 10, 100, 955, 9296, 92704, 944636, 9801929, 103262436, 1101802764, 11883775540, 129365990061, 1419569592748, 15686292728288, 174399501150236, 1949516926153045, 21898270953801720, 247045453792464294, 2797968888077323968, 31801559116255638374, 362622937212800684560
Offset: 0

Views

Author

Andrei Asinowski, May 13 2020

Keywords

Comments

A 4_3-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -3, and ends at the x-axis.

Examples

			For n = 1, a(1) = 10 is the total number of down-steps after the last up-step in Udddd, dUddd, ddUdd, dddUd.

Links

Crossrefs

Cf. A334787, A334719, A334610, A334611.

Programs

  • Mathematica
    a[n_] := 4 * Binomial[5*n + 9, n + 1]/(5*n + 9) - 16 * Binomial[5*n + 4, n]/(5*n + 4); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

Formula

a(n) = 4*binomial(5*(n+1)+4, n+1)/(5*(n+1)+4) - 16*binomial(5*n+4, n)/(5*n+4).
G.f.: (4 - 21*x - 4*(1 - 4*x)*HypergeometricPFQ([-1/5, 1/5, 2/5, 3/5], [1/4, 1/2, 3/4], 3125*x/256))/(5*x^2). - Stefano Spezia, Apr 25 2023

A334651 a(n) is the total number of down steps between the first and second up steps in all 4_1-Dyck paths of length 5*n.

Original entry on oeis.org

0, 7, 25, 155, 1195, 10282, 94591, 910480, 9054965, 92310075, 959473878, 10129715890, 108327387675, 1170975480360, 12773887368040, 140445927510832, 1554748206904325, 17314584431331025, 193849445090545875, 2180550929942519685, 24632294533221865028
Offset: 0

Views

Author

Benjamin Hackl, May 13 2020

Keywords

Comments

A 4_1-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 = -1.
For n = 1, there is no 2nd up step, a(1) = 7 enumerates the total number of down steps between the 1st up step and the end of the path.

Examples

			For n = 1, the 4_1-Dyck paths are DUDDD, UDDDD. This corresponds to a(1) = 3 + 4 = 7 down steps between the 1st up step and the end of the path.

Links

Crossrefs

Cf. A002294, A124724, A334642, A334647, A334719, A334786, A334787.

Programs

  • Mathematica
    a[0] = 0; a[n_] := 4 * Binomial[5*n, n]/(n + 1) - 3 * Binomial[5*n + 1, n]/(n + 1) + 8*Binomial[5*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)
  • SageMath
    [4*binomial(5*n, n)/(n + 1) - 3*binomial(5*n + 1, n)/(n + 1) + 8*binomial(5*(n - 1), n - 1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)]

Formula

a(0) = 0 and a(n) = 4*binomial(5*n, n)/(n+1) - 3*binomial(5*n+1, n)/(n+1) + 8*binomial(5*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.
Showing 1-4 of 4 results.

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