A334979
a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 3_1-Dyck paths of length 4*n. 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.
Original entry on oeis.org
0, 1, 16, 132, 1034, 8134, 64880, 525132, 4307512, 35750473, 299759200, 2535849836, 21619615164, 185582339740, 1602675301920, 13915031036412, 121396437548136, 1063653520870612, 9355905795325888, 82585983533819920, 731350409249262330, 6495673923406863630
Offset: 0
For n = 2, the 3_1-Dyck paths are UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD, DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 4 + 3 + 2 + 1 + 0 + 3 + 2 + 1 + 0 = 16.
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a[0] = 0; a[n_] := Binomial[4*n+6, n+1]/(2*n + 3) - 4 * Binomial[4*n + 2, n]/(2*n + 1); Array[a, 22, 0]
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[binomial(4*n + 6, n + 1)/(2*n + 3) - 4*binomial(4*n + 2, n)/(2*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020
A334976
a(n) is the total number of down steps between the (n-1)-th and n-th up steps in all 2-Dyck paths of length 3*n. A 2-Dyck path is a nonnegative path with steps (1, 2), (1, -1) that starts and ends at y = 0.
Original entry on oeis.org
0, 0, 3, 19, 108, 609, 3468, 20007, 116886, 690690, 4122495, 24823188, 150629248, 920274804, 5656456104, 34954487967, 217044280458, 1353539406660, 8474029162305, 53241343026795, 335592121524660, 2121577490385885, 13448859209014320, 85467026778421860
Offset: 0
For n = 2, the 2-Dyck paths are UDDUDD, UDUDDD, UUDDDD. Therefore the total number of down steps between the first and second up steps is a(2) = 2+1+0 = 3.
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alias(PS=ListTools:-PartialSums): A334976List := proc(m) local A, P, n;
A := [0,0]; P := [1,0]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A334976List(24); # Peter Luschny, Mar 26 2022
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a[0] = 0; a[n_] := Binomial[3*n+4, n+1]/(3*n + 4) - 3 * Binomial[3*n + 1, n]/(3*n + 1); Array[a, 24, 0]
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[binomial(3*n + 4, n + 1)/(3*n + 4) - 3*binomial(3*n + 1, n)/(3*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020
A334977
a(n) is the total number of down steps between the (n-1)-th and n-th 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, 1, 9, 53, 299, 1692, 9690, 56221, 330165, 1959945, 11745435, 70974252, 432019844, 2646716264, 16307880462, 100996570221, 628356589721, 3925544432355, 24616047166095, 154886752443885, 977595783524955, 6187863825170160, 39269844955755960, 249819662230403148
Offset: 0
For n = 2, the 2_1-Dyck paths are UDDDUD, UDDUDD, UDUDDD, UUDDDD, DUDDUD, DUDUDD, DUUDDD. Therefore the total number of down steps between the first and second up step is a(2) = 3 + 2 + 1 + 0 + 2 + 1 +0 = 9.
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a[0] = 0; a[n_] := 2*Binomial[3*n+5, n+1]/(3*n + 5) - 6 * Binomial[3*n + 2, n]/(3*n + 2); Array[a, 24, 0]
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[2*binomial(3*n + 5, n + 1)/(3*n + 5) - 6*binomial(3*n + 2, n)/(3*n + 2) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020
A334978
a(n) is the total number of down steps between the (n-1)-th and n-th 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, 6, 52, 409, 3208, 25484, 205452, 1679332, 13894848, 116193246, 980658172, 8343605534, 71492410640, 616418176920, 5344364518140, 46565472754044, 407529832131712, 3580911446989368, 31579384975219920, 279414033129153065, 2479725948121016040
Offset: 0
For n = 2, the 3-Dyck paths are UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. Therefore, the total number of down steps between the first and second up steps is a(2) = 3 + 2 + 1 + 0 = 6.
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a[0] = 0; a[n_] := Binomial[4*n+5, n+1]/(4*n + 5) - 4 * Binomial[4*n + 1, n]/(4*n + 1); Array[a, 22, 0]
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[binomial(4*n + 5, n + 1)/(4*n + 5) - 4*binomial(4*n + 1, n)/(4*n + 1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 19 2020
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