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) = A247294(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- z^2 ).
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) +4*(-n+6)*a(n-6) +(-2*n+15)*a(n-7) +(n-9)*a(n-8) +(2*n-21)*a(n-9) +(n-12)*a(n-10)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(6)=30 because among the 37 (=A004148(7)) members of B(6) only uhdhh, huhdh, hhuhd, Huhd, uhdH, uHdh, and huHd contain uhd or uHd (or both).
MAPLE
eq := G = 1+z*G+z^2*G+z^3*(G-z-z^2)*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-1, `if`(t=2, 3, 0))+b(n-1, y,
`if`(t=1, 2, 0))+`if`(n>1, b(n-2, y, `if`(t=1, 2, 0))+
b(n-2, y+1, 1), 0)))
end:
a:= n-> b(n, 0$2):
seq(a(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-1, If[t == 2, 3, 0]] + b[n-1, y, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, If[t == 1, 2, 0]] + b[n-2, y+1, 1], 0]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 16 2014
STATUS
approved