This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A182885 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2. These are paths 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. 0
 1, 1, 1, 1, 3, 2, 7, 3, 1, 13, 10, 3, 27, 29, 6, 1, 61, 66, 22, 4, 133, 157, 75, 10, 1, 287, 398, 201, 40, 5, 633, 975, 538, 155, 15, 1, 1407, 2334, 1506, 476, 65, 6, 3121, 5631, 4077, 1414, 280, 21, 1, 6943, 13602, 10695, 4320, 966, 98, 7, 15517, 32623, 27966, 12765, 3150, 462, 28, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Sum of entries in row n is A051286(n). T(n,0)=A098479(n). Sum(k*T(n,k), k=0..n)=A182886(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 FORMULA G.f.: G(t,z) =1/sqrt(1-2z-2tz^2+z^2+2t*z^3+t^2*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; 1; 1,1; 3,2; 7,3,1; 13,10,3; 27,29,6,1; MAPLE G:=1/sqrt(1-2*z-2*t*z^2+z^2+2*t*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, k), k=0..floor(n/2)) od; # yields sequence in triangular form CROSSREFS Cf. A051286, A098479, A182886. Sequence in context: A175920 A200593 A099378 * A182891 A071190 A057020 Adjacent sequences:  A182882 A182883 A182884 * A182886 A182887 A182888 KEYWORD nonn,tabf 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 | Transforms | Puzzles | Hot | Classics
Recent Additions | More pages | Superseeker | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .