A190169 Number of (1,0)-steps at levels 1,3,5,... in all peakless Motzkin paths of length n.

0, 0, 0, 1, 4, 10, 24, 60, 152, 386, 980, 2488, 6324, 16098, 41032, 104711, 267512, 684138, 1751316, 4487217, 11506792, 29530524, 75841152, 194910254, 501234960, 1289755668, 3320603016, 8553723949, 22044934324, 56841474482, 146626826376, 378392593206, 976884539336, 2522936490418

Offset

0,5

Comments

a(n)=Sum(k*A190167(n,k),k>=0).

a(n)=A110236(n) - A190166(n).

Links

Table of n, a(n) for n=0..33.

Formula

G.f. = (1-2z+z^2-2z^3+z^4)/[2z(1-z+z^2)sqrt((1+z+z^2)(1-3z+z^2))]-1/(2z).

Conjecture: -(n-1)*(n+1)*a(n) -n*(n-19)*a(n-1) +2*(n-1)*(7*n-40)*a(n-2) -(n-2)*(17*n-97)*a(n-3) +2*(9*n^2-64*n+119)*a(n-4) -17*(n-4)*(n-5)*a(n-5) +(19*n-59)*(n-5)*a(n-6) -2*(8*n-21)*(n-6)*a(n-7) +2*(2*n-5)*(n-7)*a(n-8)=0. - R. J. Mathar, Apr 09 2019

Example

a(4)=4 because in hhhh, huh'd, uh'dh, and uh'h'd, where u=(1,1), h=(1,0), d=(1,-1), we have 0+1+1+2 h-steps at odd levels (marked).

Maple

G := ((1-2*z+z^2-2*z^3+z^4)*1/2)/(z*(1-z+z^2)*sqrt((1+z+z^2)*(1-3*z+z^2)))-(1/2)/z: Gser:=series(G, z=0, 36): seq(coeff(Gser, z, n), n=0..33);

Crossrefs

Cf. A190167, A110236, A190166

Sequence in context: A080628 A225127 A230954 * A212330 A291412 A366645

Adjacent sequences: A190166 A190167 A190168 * A190170 A190171 A190172

Keyword

nonn

Author

Emeric Deutsch, May 06 2011

Status

approved