A187258 Number of UH^jD's for some j>0, in all peakless Motzkin paths of length n (here U=(1,1), D=(1,-1) and H=(1,0); can be easily expressed using RNA secondary structure terminology).
0, 0, 0, 1, 3, 7, 17, 41, 99, 242, 596, 1477, 3681, 9215, 23155, 58368, 147530, 373768, 948882, 2413264, 6147414, 15682008, 40056238, 102434119, 262228051, 671945055, 1723350315, 4423518544, 11362907022, 29208834520, 75131251334, 193370093508, 497969663062
Offset: 0
Keywords
Examples
a(4)=3 because in HHHH, HUHD, UHDH and UHHD we have 0+1+1+1 subwords of the type UH^jD.
Programs
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Maple
eq := g = 1+z*g+z^2*g*(g-1): G := RootOf(eq, g): Gser := series(z^3*G^2/((1-z)*(1-z^2*G^2)), z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
Formula
G.f.: z^3*G^2/((1-z)*(1-z^2*G^2)), where G = 1+z*G+z^2*G*(G-1).
a(n) = Sum_{k>=0} k*A089741(n,k).
D-finite with recurrence (-n+1)*a(n) +(4*n-7)*a(n-1) +(-5*n+16)*a(n-2) +(5*n-22)*a(n-3) +(-5*n+18)*a(n-4) +(5*n-24)*a(n-5) +(-4*n+25)*a(n-6) +(n-7)*a(n-7)=0. - R. J. Mathar, Jul 22 2022
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