login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A182882 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 1. L_n is the set of lattice paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1; an (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. 2
1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 1, 6, 3, 0, 1, 6, 3, 12, 4, 0, 1, 7, 24, 6, 20, 5, 0, 1, 12, 34, 60, 10, 30, 6, 0, 1, 31, 60, 100, 120, 15, 42, 7, 0, 1, 40, 185, 180, 230, 210, 21, 56, 8, 0, 1, 91, 260, 645, 420, 455, 336, 28, 72, 9, 0, 1, 170, 636, 980, 1715, 840, 812, 504, 36, 90, 10, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Sum of entries in row n is A051286(n).

T(n,0)=A182883(n).

Sum(k*T(n,k), k=0..n)=A182884(n).

REFERENCES

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.

LINKS

Table of n, a(n) for n=0..77.

FORMULA

G.f.: G(t,z) =1/sqrt(1-2tz-2z^2+t^2*z^2+2t*z^3+z^4-4z^3).

EXAMPLE

T(3,1)=2. 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; two of them have exactly one h step.

Triangle starts:

1;

0,1;

1,0,1;

2,2,0,1;

1,6,3,0,1;

6,3,12,4,0,1

MAPLE

G:=1/sqrt(1-2*t*z-2*z^2+t^2*z^2+2*t*z^3+z^4-4*z^3): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 11 do seq(coeff(P[n], t, k), k=0..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A051286, A182883, A182884.

Sequence in context: A264909 A104579 A079531 * A134178 A059018 A249371

Adjacent sequences:  A182879 A182880 A182881 * A182883 A182884 A182885

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 11 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 | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 12 22:25 EDT 2021. Contains 343829 sequences. (Running on oeis4.)