A334682 -id:A334682 - cp's OEIS frontend

A334645 a(n) is the total number of down steps between the 2nd and 3rd 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.

0, 0, 18, 52, 277, 1752, 12120, 88692, 674751, 5282160, 42267384, 344152080, 2842055359, 23746693240, 200383750632, 1705243729560, 14617677294675, 126106202849760, 1094034474058488, 9538676631305712, 83536778390997780, 734521734171474400, 6481894477750488160

Offset: 0

Benjamin Hackl, May 12 2020

For n = 2, there is no 3rd up step, a(2) = 18 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 4 + 5 + 6 = 18 down steps between the 2nd up step and the end of the path.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334640, A334643, A334682, A334785.

[3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 3)]) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = 3*Sum_{j=0..2} binomial(4*j+1, j) * binomial(4*(n-j), n-j)/((4*j+1) * (n-j+1)) for n > 1.

A334647 a(n) is the total number of down steps between the first and second up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 5, 16, 78, 470, 3153, 22588, 169188, 1308762, 10374460, 83829856, 687929086, 5717602930, 48030047206, 407142435000, 3478286028840, 29917720938690, 258866494630164, 2251694583485824, 19677972159742360, 172694287830500440, 1521328368800877065

Offset: 0

Benjamin Hackl, May 12 2020

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.

For n = 1, there is no 2nd up step, a(1) = 5 enumerates the total number of down steps between the 1st up step and the end of the path.

			For n = 1, the 3_1-Dyck paths are UDDD, DUDD. This corresponds to a(1) = 3 + 2 = 5 down steps between the 1st up step and the end of the path.
For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 3 + 2 + 1 + 0 + 4 + 3 + 2 + 1 + 0 = 16 down steps between the 1st and 2nd up step.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334648, A334649, A334680, A334682, A334785.

a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - 2 * Binomial[4*n + 1, n]/(n + 1) + 6 * Binomial[4*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*(n==1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - 2*binomial(4*n+1, n)/(n+1) + 6*binomial(4*(n-1), n-1)/n - 2*[n=1] for n > 0, where [ ] is the Iverson bracket.

A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).

Original entry on oeis.org

0, 4, 30, 250, 2245, 21221, 208129, 2098565, 21619910, 226593015, 2408424760, 25899375645, 281273231985, 3080585212120, 33986840371400, 377364606387005, 4213620859310140, 47284625533425750, 532996618440511710, 6032169040263819485, 68517222947120776290

Offset: 0

Andrei Asinowski, May 08 2020

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

			For n = 2, the a(2) = 30 is the total number of down-steps after the last up-step in UddddUdddd, UdddUddddd, UddUdddddd, UdUddddddd, UUdddddddd (thus, 4 + 5 + 6 + 7 + 8).

Stefano Spezia, Table of n, a(n) for n = 0..900
Andrei Asinowski, Benjamin Hackl, and Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A002294. Cf. A062985.

Cf. A334682 (similar for 3-Dyck paths).

b:= proc(x, y) option remember; `if`(x=y, x,
     `if`(y+40, b(x-1, y-1), 0))
    end:
a:= n-> b(5*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n, (5*(5*n-4)*
      (5*n-3)*(5*n-2)*(5*n-1)*n*(2869*n^3+5354*n^2+3239*n+634)*
       a(n-1))/(8*(n-1)*(4*n+3)*(2*n+1)*(4*n+5)*(n+1)*
       (2869*n^3-3253*n^2+1138*n-120)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

a[n_] := Binomial[5*n + 6, n + 1]/(5*n + 6) - Binomial[5*n + 1, n]/(5*n + 1); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = {binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1)} \\ Andrew Howroyd, May 08 2020

[binomial(5*(n + 1) + 1, n + 1)/(5*(n + 1) + 1) - binomial(5*n + 1, n)/(5*n + 1) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1).
a(n) = A062985(n+1, 4*n-1).
G.f.: ((1 - x)*HypergeometricPFQ([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

Original entry on oeis.org

0, 0, 9, 19, 72, 324, 1595, 8307, 44982, 250648, 1427679, 8274825, 48644310, 289334160, 1738043892, 10529070020, 64252519830, 394601627376, 2437058926871, 15126463230165, 94306717535940, 590318477063700, 3708527622652755, 23374587898663155, 147770791807427880

Offset: 0

Benjamin Hackl, May 07 2020

For n = 2, there is no 3rd up step, a(2) = 9 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, there are the 2-Dyck paths UUDDDD, UDUDDD, UDDUDD. Between the 2nd up step and the end of the path there are a(2) = 4 + 3 + 2 = 9 down steps in total.

Alois P. Heinz, Table of n, a(n) for n = 0..1212
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A007226, A007228, A124724, A334641, A334642, A334682.

b:= proc(x, y, u, c) option remember; `if`(x=0, c,
     `if`(y+20, b(x-1, y-1, u, c+`if`(u=2, 1, 0)), 0))
    end:
a:= n-> b(3*n, 0$3):
seq(a(n), n=0..24);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [0$2, 9][n+1],
     (3*(n-1)*(3*n-8)*(3*n-7)*(13*n-20)*a(n-1))/
     (2*(13*n-33)*(n-2)*(2*n-3)*n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, May 09 2020

a[0] = a[1] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 2}]; Array[a, 25, 0] (* Amiram Eldar, May 09 2020 *)

a(n) = if (n<=1, 0, 2*sum(j=1, 2, binomial(3*j+1,j) * binomial(3*(n-j),n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 2020

a(0) = a(1) = 0 and a(n) = 2*Sum_{j=1..2} binomial(3*j+1,j) * binomial(3*(n-j),n-j) / ((3*j+1)*(n-j+1)) for n > 1.

A334646 a(n) is the total number of down steps between the 3rd and 4th up steps in all 3-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 0, 118, 409, 2368, 15750, 112716, 845295, 6551208, 52035714, 421286280, 3463401007, 28832656408, 242565115858, 2058945519936, 17611312647075, 151647023490480, 1313460091978458, 11435310622320552, 100019000856225156, 878443730199290560

Offset: 0

Benjamin Hackl, May 12 2020

A 3-Dyck path is a nonnegative lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0.

			For n = 3, there is no 4th up step, a(3) = 118 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334641, A334645, A334682, A334785.

a[0] = a[1] = a[2] = 0; a[n_] := 3 * Sum[Binomial[4*j + 1, j] * Binomial[4*(n - j), n - j]/((4*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[3*sum([binomial(4*j + 1, j)*binomial(4*(n - j), n - j)/(4*j + 1)/(n - j + 1) for j in srange(1, 4)]) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = a(2) = 0 and a(n) = 3*Sum_{j=1..3} binomial(4*j+1, j)*binomial(4*(n-j), n-j)/((4*j+1)*(n-j+1)) for n > 2.

A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 34, 132, 722, 4638, 32416, 238956, 1827918, 14370595, 115384756, 942115942, 7798224226, 65286060253, 551838621972, 4702955036640, 40366238473530, 348631520142879, 3027590307082804, 26420699531880832, 231571468023697960, 2037650653547067005

Offset: 0

Benjamin Hackl, May 12 2020

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.

For n = 2, there is no 3rd up step, a(2) = 34 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 34 down steps between the 2nd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334649, A334680, A334682, A334785.

a[0] = a[1] = 0; a[n_] := Binomial[4*n + 1, n]/(4*n + 1) + 6 * Sum[Binomial[4*j + 2, j] * Binomial[4*(n - j), n - j]/((4*j + 2)*(n - j + 1)), {j, 1, 2}] - 9 * Boole[n == 2]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 9*(n==2) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..2} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 9*[n=2] for n > 1, where [ ] is the Iverson bracket.

A334649 a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 0, 236, 1034, 6094, 40996, 295740, 2231022, 17370163, 138473536, 1124433142, 9266859394, 77307427741, 651540030688, 5538977450256, 47442103851930, 409000732566399, 3546232676711824, 30903652601552272, 270529448396053576, 2377829916885541565

Offset: 0

Benjamin Hackl, May 12 2020

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.

For n = 3, there is no 4th up step, a(3) = 236 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334648, A334680, A334682, A334785.

[binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 4)]) - 52*(n==3) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = a(2) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..3} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 52*[n=3] for n > 2, where [ ] is the Iverson bracket.

A334609 a(n) is the total number of down-steps after the final up-step in all 3_2-Dyck paths of length 4n (n up-steps and 3n down-steps).

Original entry on oeis.org

0, 6, 46, 339, 2553, 19723, 155805, 1253931, 10249096, 84864051, 710429304, 6003238901, 51140131770, 438729741450, 3787208722815, 32871470376123, 286706337100656, 2511620756461504, 22089299382478728, 194966351598215340, 1726424465382128205

Offset: 0

Andrei Asinowski, May 13 2020

A 3_2-Dyck path is a lattice path with steps U = (1, 3), d = (1, -1) that starts at (0,0), stays (weakly) above y = -2, and ends at the x-axis.

			For n = 1, a(1) = 6 is the total number of down-steps after the last up-step in Uddd, dUdd, ddUd.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A334785, A334650, A334682, A334608.

a[n_] := 3 * Binomial[4*n + 7, n + 1]/(4*n + 7) - 9 * Binomial[4*n + 3, n]/(4*n + 3); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

[3*binomial(4*(n + 1) + 3, n + 1)/(4*(n + 1) + 3) - 9*binomial(4*n + 3, n)/(4*n + 3) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = 3*binomial(4*(n+1) + 3, n+1)/(4*(n+1) + 3) - 9*binomial(4*n+3, n)/(4*n + 3).

A334641 a(n) is the total number of down steps between the 3rd and 4th up steps in all 2-Dyck paths of length 3*n.

Original entry on oeis.org

0, 0, 0, 43, 108, 444, 2099, 10683, 56994, 314296, 1776519, 10236081, 59892690, 354886920, 2125117332, 12839859620, 78176677734, 479177993904, 2954360065247, 18309779343549, 114001476318240, 712751759478780, 4472908385838795, 28165267333869435

Offset: 0

Benjamin Hackl, May 07 2020

A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

For n = 3, there is no 4th up step, a(3) = 43 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A007226, A007228, A124724, A334640, A334642, A334682.

a[0] = a[1] = a[2] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 24, 0] (* Amiram Eldar, May 09 2020 *)

a(n) = if (n<=2, 0, 2*sum(j=1, 3, binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 2020

a(0) = a(1) = a(2) = 0 and a(n) = 2*Sum_{j=1..3}binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)) for n > 2.

A334608 a(n) is the total number of down-steps after the final up-step in all 3_1-Dyck paths of length 4*n (n up-steps and 3n down-steps).

Original entry on oeis.org

0, 5, 34, 236, 1714, 12922, 100300, 796572, 6443536, 52909593, 439896626, 3695917940, 31331587252, 267669458420, 2302188456120, 19918434257052, 173240112503520, 1513821095788420, 13283883136738344, 117009704490121520, 1034217260142108570, 9169842145476773250, 81537271617856588380

Offset: 0

Andrei Asinowski, May 13 2020

A 3_1-Dyck path is a lattice path with steps U=(1, 3), d=(1, -1) that starts at (0,0), stays (weakly) above y=-1, and ends at the x-axis.

			For n=1, a(1)=5 is the total number of down-steps after the last up-step in Uddd, dUdd.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334609, A334645, A334646, A334647, A334648, A334649, A334680, A334682, A334785.

a[n_] := 2 * Binomial[4*n + 6, n + 1]/(4*n + 6) - 4 * Binomial[4*n + 2, n]/(4*n + 2); Array[a, 23, 0] (* Amiram Eldar, May 13 2020 *)

[2*binomial(4*(n + 1) + 2, n + 1)/(4*(n + 1) + 2) - 4*binomial(4*n + 2, n)/(4*n + 2) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = 2*binomial(4*(n+1)+2, n+1)/(4*(n+1)+2) - 4*binomial(4*n+2, n)/(4*n+2).

cp's OEIS Frontend

A334645 a(n) is the total number of down steps between the 2nd and 3rd 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.

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A334647 a(n) is the total number of down steps between the first and second up steps in all 3_1-Dyck paths of length 4*n.

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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5*n (n up-steps and 4*n down-steps).

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A334640 a(n) is the total number of down steps between the 2nd and 3rd up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

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A334646 a(n) is the total number of down steps between the 3rd and 4th up steps in all 3-Dyck paths of length 4*n.

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A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

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A334649 a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n.

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A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).

A334609 a(n) is the total number of down-steps after the final up-step in all 3_2-Dyck paths of length 4n (n up-steps and 3n down-steps).