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

0,7

Comments

a(n)=Sum(k*A098056(n,k), k>=0).

Links

Table of n, a(n) for 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

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);

Crossrefs

Cf. A098056, A004148

Sequence in context: A350952 A000175 A276659 * A296593 A125064 A361493

Adjacent sequences: A187256 A187257 A187258 * A187260 A187261 A187262

Keyword

nonn

Author

Emeric Deutsch, May 05 2011

Status

approved