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A182890 Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n. 3
0, 1, 2, 5, 14, 36, 94, 247, 646, 1691, 4428, 11592, 30348, 79453, 208010, 544577, 1425722, 3732588, 9772042, 25583539, 66978574, 175352183, 459077976, 1201881744, 3146567256, 8237820025, 21566892818, 56462858429, 147821682470, 387002188980, 1013184884470 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The members of L_n are paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

LINKS

Table of n, a(n) for n=0..30.

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

E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).

FORMULA

G.f: x/((1+x+x^2)*(1-3*x+x^2)).

a(n) = Sum_{k>=0} k*A182888(n,k).

a(n) = (A000045(2n+2) - ((-1)^n)*A010892(n))/4. - John M. Campbell, Dec 30 2016

EXAMPLE

a(3)=5. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 0+0+1+1+3=5 h-steps at level 0.

MAPLE

G:=z/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G, z=0, 33): seq(coeff(Gser, z, n), n=0..30);

CROSSREFS

Cf. A182888.

Sequence in context: A244061 A297120 A299167 * A102714 A087223 A005955

Adjacent sequences:  A182887 A182888 A182889 * A182891 A182892 A182893

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Dec 12 2010

STATUS

approved

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Last modified September 20 01:04 EDT 2020. Contains 337228 sequences. (Running on oeis4.)