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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.

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A334610 a(n) is the total number of down-steps after the final up-step in all 4_1-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

Original entry on oeis.org

0, 7, 58, 505, 4650, 44677, 443238, 4507461, 46744100, 492492330, 5257084420, 56734340091, 618001356458, 6785943435960, 75033214770640, 834733624099485, 9336542892778440, 104932793226255165, 1184421713336050590, 13421053387405062290, 152613573227667516580
Offset: 0

Views

Author

Andrei Asinowski, May 13 2020

Keywords

Comments

A 4_1-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above y = -1, and ends at the x-axis.

Examples

			For n=1, a(1) = 7 is the total number of down-steps after the last up-step in Udddd, dUddd.

Links

Crossrefs

Cf. A002294, A334651, A334719, A334611, A334612.

Programs

  • Mathematica
    a[n_] := 2 * Binomial[5*n + 7, n + 1]/(5*n + 7) - 4 * Binomial[5*n + 2, n]/(5*n + 2); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

Formula

a(n) = 2*binomial(5*(n+1)+2, n+1)/(5*(n+1)+2) - 4*binomial(5*n+2, n)/(5*n+2).
G.f.: ((1 - 2*x)*HypergeometricPFQ([2/5, 3/5, 4/5, 6/5], [3/4, 5/4, 3/2], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

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
Showing 1-2 of 2 results.

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