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A182878 Triangle read by rows: T(n,k) is the number of lattice paths L_n of weight n having length k (0 <= k <= n). These are paths 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. 2
1, 0, 1, 0, 1, 1, 0, 0, 4, 1, 0, 0, 1, 9, 1, 0, 0, 0, 9, 16, 1, 0, 0, 0, 1, 36, 25, 1, 0, 0, 0, 0, 16, 100, 36, 1, 0, 0, 0, 0, 1, 100, 225, 49, 1, 0, 0, 0, 0, 0, 25, 400, 441, 64, 1, 0, 0, 0, 0, 0, 1, 225, 1225, 784, 81, 1, 0, 0, 0, 0, 0, 0, 36, 1225, 3136, 1296, 100, 1, 0, 0, 0, 0, 0, 0, 1, 441, 4900, 7056, 2025, 121, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

The weight of a path is the sum of the weights of its steps.

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

Sum_{k=0..n} k*T(n,k) = A182879(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..90.

FORMULA

T(n,k) = binomial(n,n-k)^2.

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

EXAMPLE

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 hhh, hH, Hh, ud, and du, having lengths 3, 2, 2, 2, and 2, respectively.

Triangle starts:

  1;

  0,  1;

  0,  1,  1;

  0,  0,  4,  1;

  0,  0,  1,  9,  1;

  0,  0,  0,  9, 16,  1;

MAPLE

T:=(n, k)->binomial(k, n-k)^2: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A051286, A182879.

Sequence in context: A036877 A049763 A328290 * A221971 A297785 A085992

Adjacent sequences:  A182875 A182876 A182877 * A182879 A182880 A182881

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 10 2010

EXTENSIONS

Keyword tabl added by Michel Marcus, Apr 09 2013

STATUS

approved

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Last modified December 10 20:36 EST 2019. Contains 329909 sequences. (Running on oeis4.)