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A247291 Number of weighted lattice paths B(n) having no uhd strings. 4
1, 1, 2, 4, 7, 15, 32, 69, 154, 346, 786, 1806, 4180, 9745, 22865, 53938, 127865, 304447, 727733, 1745736, 4201350, 10140975, 24544000, 59551327, 144822097, 352940719, 861839226, 2108381480, 5166749329, 12681855551, 31174671514, 76742344774 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.

a(n) = A247290(n,0).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

FORMULA

G.f. G = G(z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z).

D-finite with recurrence +(n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +3*(n-3)*a(n-4) +(-2*n+9)*a(n-5) +2*(-n+6)*a(n-6) +(n-9)*a(n-8)=0. - R. J. Mathar, Sep 29 2021

EXAMPLE

a(4)=7 because we have hhhh, hhH, hHh, Hhh, HH, hud, and udh.

MAPLE

eq := G = 1+z*G+z^2*G+z^3*(G-z)*G: G := RootOf(eq, G): Gser := series(G, z = 0, 37): seq(coeff(Gser, z, n), n = 0 .. 35);

# second Maple program:

b:= proc(n, y, t) option remember; `if`(y<0 or y>n or t=3, 0,

      `if`(n=0, 1, b(n-1, y, `if`(t=1, 2, 0))+`if`(n>1, b(n-2,

       y, 0)+b(n-2, y+1, 1), 0)+b(n-1, y-1, `if`(t=2, 3, 0))))

    end:

a:= n-> b(n, 0$2):

seq(T(n), n=0..40); # Alois P. Heinz, Sep 16 2014

MATHEMATICA

b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n || t == 3, 0, If[n == 0, 1, b[n-1, y, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, 0] + b[n-2, y+1, 1], 0] + b[n-1, y-1, If[t == 2, 3, 0]]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, May 27 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A247290, A247293, A247295.

Sequence in context: A129682 A129981 A072964 * A030033 A280031 A135131

Adjacent sequences:  A247288 A247289 A247290 * A247292 A247293 A247294

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 16 2014

STATUS

approved

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Last modified October 25 03:56 EDT 2021. Contains 348237 sequences. (Running on oeis4.)