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A129158 Number of primitive non-Dyck factors in all skew Dyck paths of semilength n. 3
0, 0, 1, 5, 22, 96, 422, 1871, 8360, 37610, 170222, 774561, 3541487, 16263250, 74981226, 346957923, 1610847944, 7501970397, 35038158569, 164083453482, 770312822822, 3624741537711, 17093452878067, 80773023036909 (list; graph; refs; listen; history; text; internal format)



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 non-Dyck factor is a subpath of the form UPD, P being a skew Dyck path with at least one L step, or of the form UPL, P being any nonempty skew Dyck path.


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


a(n) = Sum_{k=0,..,n} k*A129157(n,k).

a(n) = A128742(n) - A129156(n).

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

a(n) ~ (3*sqrt(5)+5) * 5^(1+n) / (36*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014


a(2)=1 because in all skew Dyck paths of semilength 3, namely UDUD, UUDD and (UUDL), we have altogether 1 primitive non-Dyck factor (shown between parentheses).


G:=(1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2)-sqrt((1-4*z)*(1-6*z+5*z^2)))/(1+z+sqrt(1-6*z+5*z^2))^2: Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..27);


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


(PARI) z='z+O('z^25); concat([0, 0], Vec((1-5*z+3*(1-z)*sqrt(1-4*z)-3*sqrt(1-6*z+5*z^2) - sqrt((1-4*z)*(1-6*z+5*z^2))) /(1+z+ sqrt(1-6*z+5*z^2) )^2)) \\ G. C. Greubel, Feb 09 2017


Cf. A129157, A129156, A128742.

Sequence in context: A083586 A200676 A297333 * A129164 A123347 A087439

Adjacent sequences:  A129155 A129156 A129157 * A129159 A129160 A129161




Emeric Deutsch, Apr 02 2007



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Last modified December 3 16:20 EST 2020. Contains 338906 sequences. (Running on oeis4.)