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

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

A334611 a(n) is the total number of down-steps after the final up-step in all 4_2-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

Original entry on oeis.org

0, 9, 82, 747, 7065, 69098, 694272, 7127865, 74468546, 789265125, 8466019380, 91736269053, 1002710879409, 11042713886256, 122413333216960, 1364880618458565, 15296452128008100, 172218124701600741, 1946960139291303222, 22092883135853433030, 251545025683283255770
Offset: 0

Views

Author

Andrei Asinowski, May 13 2020

Keywords

Comments

A 4_2-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.

Examples

			For n=1, a(1) = 9 is the total number of down-steps after the last up-step in Udddd, dUddd, ddUdd.

Links

Crossrefs

Cf. A334786, A334719, A334610, A334612.

Programs

  • Mathematica
    a[n_] := 3 * Binomial[5*n + 8, n + 1]/(5*n + 8) - 9 * Binomial[5*n + 3, n]/(5*n + 3); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

Formula

a(n) = 3*binomial(5*(n+1)+3, n+1)/(5*(n+1)+3) - 9*binomial(5*n+3, n)/(5*n+3).
G.f.: ((1 - 3*x)*HypergeometricPFQ([3/5, 4/5, 6/5, 7/5], [5/4, 3/2, 7/4], 3125*x/256) - 1)/x. - 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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