[seqfan] Motzkin n-paths avoiding even-numbered steps that are up steps.

[David Scambler]

Seqfans,

This is interesting because the counts are the interleaving of the closely related sequences 
A109081 and A106228.

http://oeis.org/search?q=A106228|A109081&sort=&language=english&go=Search


1,1,1,2,3,6,10,21,37,80,146,322,602,1347,2563,5798,11181,25512,49720,114236,224540,518848,
1027038,2384538,4748042,11068567,22150519,51817118,104146733,244370806,493012682,1159883685,
2347796965,5536508864,11239697816,26560581688,54061835288,127993221140,261130778516,
619280193640,1266125122956,3007251366000,6160158505040,14651743202152,30065608532008,
71601107803904,147161532388934

MMA:
f[n_,x_,y_]:=f[n,x,y] = If[x>n||y<0,0,If[x==n&&y==0,1, If[EvenQ[x],0,
If[y>n-x-1,0,f[n,x+1,y+1]]] +If[y<=0,0,f[n,x+1,y-1]] + f[n,x+1,y]]];
Table[f[n,0,0],{n,0,46}]

dave