login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A182895 Number of (1,0)-steps at level 0 in all weighted lattice paths in L_n. 3
0, 1, 3, 7, 19, 50, 130, 341, 893, 2337, 6119, 16020, 41940, 109801, 287463, 752587, 1970299, 5158310, 13504630, 35355581, 92562113, 242330757, 634430159, 1660959720, 4348449000, 11384387281, 29804712843, 78029751247, 204284540899 (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
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.
FORMULA
a(n) = Sum_{k>=0} k*A182893(n,k).
G.f.: z(1+z)/[(1+z+z^2)(1-3z+z^2)].
a(n) = (A000032(2n+1) - A010892(2n))/4. - John M. Campbell, Dec 30 2016
4*a(n) = -A057078(n) +A002878(n). - R. J. Mathar, Jul 26 2022
EXAMPLE
a(3) = 7. 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+2+2+3=7 (1,0)-steps at level 0.
MAPLE
G:=z*(1+z)/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28);
MATHEMATICA
LinearRecurrence[{2, 1, 2, -1}, {0, 1, 3, 7}, 30] (* Harvey P. Dale, Jan 05 2022 *)
CROSSREFS
Cf. A182893.
Sequence in context: A151266 A147234 A171854 * A087224 A308398 A341703
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 12 2010
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 22 08:57 EST 2024. Contains 370248 sequences. (Running on oeis4.)