Benjamin Hackl - cp's OEIS frontend

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.

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

Offset: 0

Benjamin Hackl, May 13 2020

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.

			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.

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

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

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 *)

[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)]

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.

A334650 a(n) is the total number of down steps between the first and second up steps in all 3_2-Dyck paths of length 4*n.

Original entry on oeis.org

0, 6, 31, 158, 975, 6639, 48050, 362592, 2820789, 22460120, 182141553, 1499143282, 12490923757, 105150960654, 892973346300, 7640934031920, 65813450140017, 570160918044288, 4964875184429660, 43431741548248440, 381496856026500220, 3363457643008999635

Offset: 0

Benjamin Hackl, May 13 2020

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.

For n = 1, there is no 2nd up step, a(1) = 6 enumerates the total number of down steps between the 1st up step and the end of the path.

			For n = 1, the 3_2-Dyck paths are DDUD, DUDD, UDDD. This corresponds to a(1) = 1 + 2 + 3 = 6 down steps between the 1st 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. A001764, A007226, A007228, A334609, A334647, A334648, A334649, A334785.

a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - Binomial[4*n + 2, n]/(n + 1) + 9 * Binomial[4*(n - 1), n - 1]/n - 6 * Boole[n == 1]; Array[a, 22, 0] (* Amiram Eldar, May 13 2020 *)

[3*binomial(4*n, n)/(n + 1) - binomial(4*n + 2, n)/(n + 1) + 9*binomial(4*(n - 1), n - 1)/n - 6*(n==1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 13 2020

a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) + 9*binomial(4*(n-1), n-1)/n - 6*[n=1] for n > 0, where [ ] is the Iverson bracket.

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.

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.

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.

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.

A334643 a(n) is the total number of down steps between the second and third 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, 0, 16, 53, 209, 963, 4816, 25367, 138531, 777041, 4449511, 25901655, 152818458, 911755012, 5491420104, 33343242196, 203881825163, 1254342228285, 7759025239189, 48227078649155, 301056318504165, 1886647802277315, 11864793375611820, 74854437302309175

Offset: 0

Benjamin Hackl, May 12 2020

For n = 2, there is no 3rd up step, a(2) = 16 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 4 + 3 + 2 + 1 + 1 + 2 + 3 = 16 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. A001764, A007226, A030983, A334640, A334642, A334644.

[binomial(3*n + 1, n)/(3*n + 1) + 4*sum([binomial(3*j + 2, j)*binomial(3*(n - j), n - j)/(3*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 7*(n==2) if n >= 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = binomial(3*n+1, n)/(3*n+1) + 4*Sum_{j=1..2}binomial(3*j+2, j)*binomial(3*(n-j), n-j)/((3*j+2)*(n-j+1)) - 7*[n=2] for n > 1, where [ ] is the Iverson bracket.

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.

Original entry on oeis.org

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.

A334644 a(n) is the total number of down steps between the third and fourth 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, 0, 0, 83, 299, 1263, 6076, 31307, 168561, 936161, 5321611, 30804795, 180939408, 1075636912, 6459103704, 39120216196, 238692219923, 1465783144605, 9052278085129, 56185368932615, 350293215459915, 2192731008315015, 13775745283576920, 86831135890324875

Offset: 0

Benjamin Hackl, May 12 2020

For n = 3, there is no 4th up step, a(3) = 83 enumerates the total number of down steps between the 3rd 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. A001764, A007226, A030983, A334642, A334643.

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

a(0) = a(1) = a(2) = 0 and a(n) = binomial(3*n+1, n)/(3*n+1) + 4*Sum_{j=1..3}binomial(3*j+2, j)*binomial(3*(n-j), n-j)/((3*j+2)*(n-j+1)) - 30*[n=3] for n > 2, where [ ] is the Iverson bracket.

A334642 a(n) is the total number of down steps between the first and second 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, 3, 9, 32, 139, 669, 3430, 18360, 101403, 573551, 3305445, 19340100, 114579348, 685962172, 4143459504, 25220816752, 154545611355, 952583230899, 5902090839715, 36738469359480, 229636903762035, 1440759023752125, 9070230371741490, 57278432955350880

Offset: 0

Benjamin Hackl, May 07 2020

For n = 1, there is no 2nd up step, a(1) = 3 enumerates the total number of down steps between the 1st up step and the end of the path.

			For n = 1, the 2_1-Dyck paths are UDD, DUD. This corresponds to a(1) = 2 + 1 = 3 down steps between the 1st up step and the end of the path.
For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9 down steps between the 1st and 2nd up step.

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

Cf. A007226, A007228, A124724.

a[0] = 0; a[n_] := 2 * Binomial[3*n, n]/(n + 1) - Binomial[3*n + 1, n]/(n + 1) + 4 * Binomial[3*(n - 1), n - 1]/n - 2 * Boole[n == 1]; Array[a, 24, 0] (* Amiram Eldar, May 09 2020 *)

a(n) = if (n==0, 0, 2*binomial(3*n, n)/(n+1) - binomial(3*n+1, n)/(n+1) + 4*binomial(3*(n-1), n-1)/n - 2*(n==1)); \\ Michel Marcus, May 09 2020

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

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User: Benjamin Hackl

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.

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A334650 a(n) is the total number of down steps between the first and second up steps in all 3_2-Dyck paths of length 4*n.

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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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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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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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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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A334643 a(n) is the total number of down steps between the second and third 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.

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