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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, 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) 2191-2203

FORMULA

a(n) = Sum_{k=0,..,n} k*A129154(n,k).

a(n) = A128742(n) - A129158(n).

G.f.: (3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1 +z + sqrt(1 - 6*z + 5*z^2))^2.

a(n) ~ (5-sqrt(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:=(3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*z))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);

MATHEMATICA

CoefficientList[Series[(3-3*x-Sqrt[1-6*x+5*x^2])*(1-Sqrt[1-4*x])/ (1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PROG

(PARI) z='z+O('z^25); concat([0], Vec((3-3*z-sqrt(1-6*z+5*z^2))*(1-sqrt(1-4*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

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Last modified August 19 11:07 EDT 2017. Contains 290797 sequences.