

A129156


Number of primitive Dyck factors in all skew Dyck paths of semilength n.


3



0, 1, 3, 10, 36, 136, 532, 2139, 8796, 36859, 156946, 677514, 2959669, 13063493, 58184838, 261230814, 1181144792, 5374078726, 24588562675, 113067256235, 522270436044, 2422244159067, 11275548912967, 52663412854571
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OFFSET

0,3


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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 primitive Dyck factor is a subpath of the form UPD that starts on the xaxis, P being a Dyck path.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

a(n) = Sum_{k=0,..,n} k*A129154(n,k).
a(n) = A128742(n)  A129158(n).
G.f.: (33*zsqrt(16*z+5*z^2))*(1sqrt(14*z))/(1 +z + sqrt(1  6*z + 5*z^2))^2.
a(n) ~ (5sqrt(5)) * 5^(n+3/2) / (36*sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Mar 20 2014


EXAMPLE

a(2)=3 because in all skew Dyck paths of semilength 3, namely (UD)(UD), (UUDD) and UUDL, we have altogether 3 primitive Dyck factors (shown between parentheses).


MAPLE

G:=(33*zsqrt(16*z+5*z^2))*(1sqrt(14*z))/(1+z+sqrt(16*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);


MATHEMATICA

CoefficientList[Series[(33*xSqrt[16*x+5*x^2])*(1Sqrt[14*x])/ (1+x+Sqrt[16*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG

(PARI) z='z+O('z^25); concat([0], Vec((33*zsqrt(16*z+5*z^2))*(1sqrt(14*z))/(1 +z + sqrt(1  6*z + 5*z^2))^2)) \\ G. C. Greubel, Feb 09 2017


CROSSREFS

Cf. A129154, A129157, A129158.
Sequence in context: A007582 A026854 A136576 * A171753 A002212 A149041
Adjacent sequences: A129153 A129154 A129155 * A129157 A129158 A129159


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Apr 02 2007


STATUS

approved



