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


%S 0,1,2,3,6,14,34,83,202,495,1224,3046,7616,19115,48130,121527,307602,

%T 780244,1982834,5047377,12867438,32847357,83952780,214806750,

%U 550170300,1410412561,3618785462,9292203549,23877482490,61397367692,157972743178,406693829059,1047585820586,2699811117189

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

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

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

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

%F 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

%e 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).

%p 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);

%Y Cf. A190164, A110236, A190169, A004148

%K nonn

%O 0,3

%A _Emeric Deutsch_, May 06 2011

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Last modified October 28 16:41 EDT 2021. Contains 348329 sequences. (Running on oeis4.)