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A182890
Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n.
13
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
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
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 and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
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
a(n) = Sum_{m=0..n} C(2*n-2*m,2*m+1)/2. - Vladimir Kruchinin, Jan 24 2022
EXAMPLE
a(3)=5. Indeed, denoting by h (resp. H) the (1,0)-step of weight 1 (resp. 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);
MATHEMATICA
Table[Sum[Binomial[2n+2-2k, 2k-1]/2, {k, 0, n+1}], {n, 0, 30}]; (* Rigoberto Florez, Apr 10 2023 *)
PROG
(Maxima) a(n):=1/2*sum(binomial(2*n-2*m, 2*m+1), m, 0, (2*n-1)/4); /* Vladimir Kruchinin, Jan 24 2022 */
CROSSREFS
Cf. A182888.
Sequence in context: A244061 A297120 A299167 * A102714 A087223 A363106
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Dec 12 2010
STATUS
approved