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A129169
Number of UU's starting at level 0 in all skew Dyck paths of semilength n.
2
0, 0, 2, 9, 37, 153, 643, 2745, 11881, 52038, 230268, 1028007, 4625013, 20949444, 95462022, 437318631, 2012933893, 9304997544, 43179620542, 201076887771, 939358727203, 4401177906249, 20676208303727, 97374640613139
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.
LINKS
G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 0..200 from Vincenzo Librandi)
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
FORMULA
a(n) = Sum_{k=0,..,floor(n/2)} k*A129168(n,k).
G.f.: 2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2))^2.
Recurrence: 2*n*(n+2)*a(n) = (13*n^2+8*n+4)*a(n-1) - (16*n^2-7*n-18)*a(n-2) + 5*(n-2)*(n+1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 5^(n+3/2)/(6*sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
a(3)=9 because in the 10 skew Dyck paths of semilength 3, namely UDUDUD, UD(UU)DD, UD(UU)DL, (UU)DDUD, (UU)DUDD, (UU)UDDD, (UU)UDLD, (UU)DUDL, (UU)UDDL and (UU)UDLL, we have altogether 9 UU's starting at level 0 (shown between parentheses).
MAPLE
G:=2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(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[2*(2-x)*(1-3*x-Sqrt[1-6*x+5*x^2])/(1+x+Sqrt[1-6*x+5*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) z='z+O('z^25); concat([0, 0], Vec(2*(2-z)*(1-3*z-sqrt(1-6*z+5*z^2))/(1+z+sqrt(1-6*z+5*z^2))^2)) \\ G. C. Greubel, Feb 09 2017
CROSSREFS
Cf. A129169.
Sequence in context: A178875 A012493 A106851 * A162548 A150983 A150984
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Apr 05 2007
STATUS
approved