A191393 Number of dispersed Dyck paths of length n (i.e., of Motzkin paths of length n with no (1,0)-steps at positive heights) having no base pyramids. A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1).

1, 1, 1, 1, 1, 1, 2, 3, 8, 13, 31, 49, 109, 170, 371, 581, 1270, 2010, 4417, 7063, 15581, 25123, 55554, 90179, 199752, 326089, 723351, 1186670, 2635764, 4342829, 9657336, 15973459, 35558165, 59017088, 131500422, 218932442, 488234057, 815127111, 1819186163

Offset

0,7

Comments

a(n) = A191392(n,0).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

a(n) ~ 9 * 2^(n-11/2) * (16+(-1)^n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

D-finite with recurrence (n+1)*a(n) +3*(-n-1)*a(n-1) +(-5*n+13)*a(n-2) +(19*n-35)*a(n-3) +3*(n-11)*a(n-4) +2*(-17*n+73)*a(n-5) +(5*n-13)*a(n-6) +2*(13*n-77)*a(n-7) +4*(-n+8)*a(n-8) +8*(-n+8)*a(n-9)=0. - R. J. Mathar, Jul 26 2022

Example

a(7)=3 because we have HHHHHHH, HUUDUDD, and UUDUDDH, where U=(1,1), D=(1,-1), and H=(1,0).

Maple

g := (2*(1-z^2))/(1-2*z+z^2+2*z^3+(1-z^2)*sqrt(1-4*z^2)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 38);

Mathematica

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

Prog

(Magma) m:=40; R<x>:=LaurentSeriesRing(RationalField(), m); Coefficients(R! (2*(1-x^2))/(1-2*x+x^2+2*x^3+(1-x^2)*Sqrt(1-4*x^2))); // Vincenzo Librandi, Mar 21 2014

Crossrefs

Cf. A191392.

Sequence in context: A045692 A103196 A113954 * A025082 A317911 A151889

Adjacent sequences: A191390 A191391 A191392 * A191394 A191395 A191396

Keyword

nonn

Author

Emeric Deutsch, Jun 04 2011

Status

approved