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A129167 Number of base pyramids in all skew Dyck paths of semilength n. 3
0, 1, 3, 9, 30, 109, 420, 1685, 6960, 29391, 126291, 550359, 2426502, 10803801, 48507843, 219377949, 998436792, 4569488371, 21016589073, 97090411019, 450314942682, 2096122733211, 9788916220518, 45850711498859, 215348942668680, 1013979873542689, 4785437476592805, 22633143884165985, 107258646298581390 (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 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.

a(n) = |A091699(n+1)|. Partial sums of A033321(n), n = 1, 2, 3, ....

a(n+1) is the number of 3-colored Motzkin paths of length n with no peaks at level 1. - José Luis Ramírez Ramírez, Mar 31 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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*A129165(n,k).

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

Recurrence: 2*(n+1)*a(n) = (13*n-3)*a(n-1) - 4*(4*n-3)*a(n-2) + 5*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 5^(n+5/2)/(72*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012

EXAMPLE

a(2)=3 because in the paths (UD)(UD), (UUDD) and UUDL we have altogether 3 base pyramids (shown between parentheses).

MAPLE

G:=(1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*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[(1-3*x-Sqrt[1-6*x+5*x^2])/(x*(3-3*x-Sqrt[1-6*x+5*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PROG

(PARI) z='z+O('z^66); concat([0], Vec((1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)))) \\ Joerg Arndt, Aug 27 2014

CROSSREFS

Cf. A033321, A091699, A129165.

Sequence in context: A032125 A246472 A091699 * A151472 A107379 A117428

Adjacent sequences:  A129164 A129165 A129166 * A129168 A129169 A129170

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Apr 04 2007

STATUS

approved

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Last modified December 12 06:05 EST 2017. Contains 295937 sequences.