A187259 Number of UH^jU's, DH^jD's, and DH^jU'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, 0, 0, 0, 2, 11, 39, 122, 358, 1008, 2770, 7493, 20049, 53239, 140603, 369837, 969883, 2537685, 6628215, 17288950, 45048932, 117285552, 305159262, 793581817, 2062948149, 5361112383, 13929080271, 36183941553, 93984332531, 244094334682, 633922350198, 1646271999611
Offset: 0
Keywords
Programs
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Maple
eq := g = 1+z*g+z^2*g*(g-1): g := RootOf(eq, g): gser := series(z^5*g^2*(3*g-1)*(g-1)/((1-z)*(1-z^2*g^2)), z = 0, 38): seq(coeff(gser, z, n), n = 0 .. 33);
Formula
G.f.=z^5*G^2*(3G-1)(G-1)/[(1-z)(1-z^2*G^2)], where G=1+zG+z^2*G(G-1).
Conjecture D-finite with recurrence -(n+1)*(42968*n-187991)*a(n) +(-33354*n^2+888062*n+187991)*a(n-1) +(587317*n^2-5596253*n+61483
17)*a(n-2) +(-549823*n^2+5720814*n-11020859)*a(n-3) +(176865*n^2-2521427*n+8169148)*a(n-4) +(-587317*n^2+6446371*n-18005842)*a(n-5)
+(592791*n^2-6850333*n+19290494)*a(n-6) -(143511*n-619655)*(n-8)*a(n-7)=0. - R. J. Mathar, Jul 22 2022
Comments