A190166 Number of (1,0)-steps at levels 0,2,4,... in all peakless Motzkin paths of length n.

0, 1, 2, 3, 6, 14, 34, 83, 202, 495, 1224, 3046, 7616, 19115, 48130, 121527, 307602, 780244, 1982834, 5047377, 12867438, 32847357, 83952780, 214806750, 550170300, 1410412561, 3618785462, 9292203549, 23877482490, 61397367692, 157972743178, 406693829059, 1047585820586, 2699811117189

Offset

0,3

Comments

a(n)=Sum(k*A190164(n,k),k>=0).

a(n)=A110236(n) - A190169(n).

Links

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

Formula

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

Conjecture: (-n+1)*a(n) +(3*n-4)*a(n-1) +2*(-n+1)*a(n-2) +3*(n-2)*a(n-3) +2*(-n+3)*a(n-4) +(3*n-8)*a(n-5) +(-n+3)*a(n-6)=0. - R. J. Mathar, Apr 09 2019

a(n) ~ phi^(2*n+2) / (4 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022

Example

a(4)=6 because in h'h'h'h', h'uhd, uhdh', and uhhd, where u=(1,1), h=(1,0), d=(1,-1), we have 4+1+1+0 h-steps at even levels (marked).

Maple

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

Crossrefs

Cf. A190164, A110236, A190169, A004148.

Sequence in context: A291401 A331875 A010357 * A238823 A002995 A093467

Adjacent sequences: A190163 A190164 A190165 * A190167 A190168 A190169

Keyword

nonn

Author

Emeric Deutsch, May 06 2011

Status

approved