This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182882 #7 Mar 12 2022 13:15:29 %S A182882 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, %T A182882 10,30,6,0,1,31,60,100,120,15,42,7,0,1,40,185,180,230,210,21,56,8,0,1, %U A182882 91,260,645,420,455,336,28,72,9,0,1,170,636,980,1715,840,812,504,36,90,10,0,1 %N 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. %C A182882 Sum of entries in row n is A051286(n). %C A182882 T(n,0)=A182883(n). %C A182882 Sum(k*T(n,k), k=0..n)=A182884(n). %D A182882 M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. %D A182882 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. %F A182882 G.f.: G(t,z) =1/sqrt(1-2tz-2z^2+t^2*z^2+2t*z^3+z^4-4z^3). %e A182882 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. %e A182882 Triangle starts: %e A182882 1; %e A182882 0,1; %e A182882 1,0,1; %e A182882 2,2,0,1; %e A182882 1,6,3,0,1; %e A182882 6,3,12,4,0,1 %p A182882 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 %Y A182882 Cf. A051286, A182883, A182884. %K A182882 nonn,tabl %O A182882 0,7 %A A182882 _Emeric Deutsch_, Dec 11 2010