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 A246181 #15 Jun 29 2015 06:51:13
%S A246181 1,0,1,1,0,1,1,2,0,1,1,3,3,0,1,3,3,6,4,0,1,3,12,6,10,5,0,1,6,14,30,10,
%T A246181 15,6,0,1,11,30,40,60,15,21,7,0,1,15,65,90,90,105,21,28,8,0,1,31,95,
%U A246181 225,210,175,168,28,36,9,0,1,50,216,350,595,420,308,252,36,45,10,0,1
%N A246181 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k (1,0)-steps of weight 1. B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: a (1,0)-step of weight 1; a (1,0)-step of weight 2; a (1,1)-step of weight 2; a (1,-1)-step of weight 1. The weight of a path is the sum of the weights of its steps.
%C A246181 Number of entries in row n is n+1.
%C A246181 Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
%C A246181 T(n,0) = A025250(n+3).
%C A246181 Sum(k*T(n,k), k>=0) = A110320(n) (n>=1).
%H A246181 Alois P. Heinz, <a href="/A246181/b246181.txt">Rows n = 0..140, flattened</a>
%H A246181 M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F A246181 G.f. G=G(t,z) satisfies G = 1 + t*z*G + z^2*G + z^3*G^2.
%e A246181 Row 3 is 1,2,0,1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the four paths of weight 3 are: ud, hH, Hh, and hhh, having 0, 1, 1, and 3 (1,0)-steps of weight 1, respectively.
%e A246181 Triangle starts:
%e A246181 1;
%e A246181 0,1;
%e A246181 1,0,1;
%e A246181 1,2,0,1;
%e A246181 1,3,3,0,1;
%e A246181 3,3,6,4,0,1;
%p A246181 eq := G = 1+t*z*G+z^2*G+z^3*G^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form
%p A246181 # second Maple program:
%p A246181 b:= proc(n, y) option remember; `if`(y<0 or y>n, 0,
%p A246181 `if`(n=0, 1, expand(b(n-1, y)*x+ `if`(n>1,
%p A246181 b(n-2, y)+b(n-2, y+1), 0) +b(n-1, y-1))))
%p A246181 end:
%p A246181 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
%p A246181 seq(T(n), n=0..12); # _Alois P. Heinz_, Aug 24 2014
%t A246181 b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y]*x + If[n>1, b[n-2, y] + b[n-2, y+1], 0] + b[n-1, y-1]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 29 2015, after _Alois P. Heinz_ *)
%Y A246181 Cf. A004148, A025250, A246177.
%K A246181 nonn,tabl
%O A246181 0,8
%A A246181 _Emeric Deutsch_, Aug 23 2014