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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). 3
0, 7, 58, 505, 4650, 44677, 443238, 4507461, 46744100, 492492330, 5257084420, 56734340091, 618001356458, 6785943435960, 75033214770640, 834733624099485, 9336542892778440, 104932793226255165, 1184421713336050590, 13421053387405062290, 152613573227667516580 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
LINKS
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 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
EXAMPLE
For n=1, a(1) = 7 is the total number of down-steps after the last up-step in Udddd, dUddd.
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 *)
CROSSREFS
Sequence in context: A015566 A238810 A362210 * A244469 A006193 A139397
KEYWORD
nonn
AUTHOR
Andrei Asinowski, May 13 2020
STATUS
approved

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Last modified July 18 00:08 EDT 2024. Contains 374377 sequences. (Running on oeis4.)