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

0,5

Comments

Same as A089742 except for the first three terms. - Georg Fischer, Oct 14 2018

Links

Table of n, a(n) for 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

Example

a(4)=3 because in HHHH, HUHD, UHDH and UHHD we have 0+1+1+1 subwords of the type UH^jD.

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);

Crossrefs

Cf. A004148, A089741, A089742.

Sequence in context: A001333 A078057 A089742 * A131721 A259855 A058351

Adjacent sequences: A187255 A187256 A187257 * A187259 A187260 A187261

Keyword

nonn

Author

Emeric Deutsch, May 05 2011

Status

approved