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

Author

Emeric Deutsch, May 06 2011

Keywords

• nonn

Comments

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

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

Examples

			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).
		

Crossrefs

Cf. A190164, A110236, A190169, A004148.

Programs

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);

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