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A191389 Number of valleys at level 0 in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0) steps at positive heights). 5
0, 0, 0, 0, 1, 2, 7, 14, 37, 74, 176, 352, 794, 1588, 3473, 6946, 14893, 29786, 63004, 126008, 263950, 527900, 1097790, 2195580, 4540386, 9080772, 18696432, 37392864, 76717268, 153434536, 313889477, 627778954, 1281220733, 2562441466, 5219170052, 10438340104 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Helmut Prodinger, Dispersed Dyck paths revisited, arXiv:2402.13026 [math.CO], 2024.
FORMULA
a(n) = Sum_{k=0..n} k*A191387(n,k).
G.f.: 2*(1-2*z^2-sqrt(1-4*z^2))/(1-2*z+sqrt(1-4*z^2))^2.
a(n) ~ 2^(n-1) * (1-3*sqrt(2)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence -(n+2)*(n-4)*a(n) +2*(n+2)*(n-4)*a(n-1) +4*(n-2)*(n-3)*a(n-2) -8*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Sep 24 2021
EXAMPLE
a(5)=2 because in HHHHH, HHHUD, HHUDH, HUDHH, HUUDD, UDHHH, UDHUD, UUDDH, HUDUD, and UDUDH only the last 2 paths have 1 valley at level 0; here U=(1,1), D=(1,-1), H=(1,0).
MAPLE
g := (2*(1-2*z^2-sqrt(1-4*z^2)))/(1-2*z+sqrt(1-4*z^2))^2: gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
MATHEMATICA
CoefficientList[Series[(2*(1-2*x^2-Sqrt[1-4*x^2]))/(1-2*x+Sqrt[1-4*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^50); concat([0, 0, 0, 0], Vec(2*(1-2*z^2 -sqrt(1-4*z^2)) /(1 - 2*z + sqrt(1-4*z^2))^2)) \\ G. C. Greubel, Feb 12 2017
CROSSREFS
Cf. A191387. Convolution square of A037952 (shifted 2 places right).
Sequence in context: A256272 A320651 A167762 * A191319 A018497 A178748
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 02 2011
STATUS
approved

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Last modified April 20 14:48 EDT 2024. Contains 371844 sequences. (Running on oeis4.)