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A129166 Number of skew Dyck paths of semilength n with no base pyramids. 2
1, 0, 1, 5, 19, 73, 292, 1203, 5065, 21697, 94274, 414514, 1840981, 8247011, 37220261, 169079113, 772489020, 3547371679, 16364309243, 75799327800, 352402156770, 1643878188646, 7691841654538, 36091803172733 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
LINKS
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = A129165(n,0).
G.f.: (1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2))).
a(n) ~ 82*5^(n+1/2)/(289*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence 6*(n+1)*a(n) +2*(-25*n+11)*a(n-1) +(131*n-229)*a(n-2) +2*(-92*n+261)*a(n-3) +2*(81*n-311)*a(n-4) +(-91*n+439)*a(n-5) +(31*n-183)*a(n-6) +5*(-n+7)*a(n-7)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(2)=1 because we have UUDL.
MAPLE
G:=(1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2))): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
MATHEMATICA
CoefficientList[Series[(1-x)*(3-3*x-Sqrt[1-6*x+5*x^2])/(2-(1-x)*(1-x-Sqrt[1-6*x+5*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
PROG
(PARI) z='z+O('z^50); Vec((1-z)*(3-3*z-sqrt(1-6*z+5*z^2))/(2-(1-z)*(1-z-sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 20 2017
CROSSREFS
Cf. A129165.
Sequence in context: A255455 A255444 A287805 * A149763 A149764 A149765
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 04 2007
STATUS
approved

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Last modified February 25 13:12 EST 2024. Contains 370330 sequences. (Running on oeis4.)