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 A182906 #14 Jan 25 2019 10:36:53
%S A182906 1,1,2,1,4,2,8,5,1,17,12,3,37,28,9,1,82,66,25,4,185,156,66,14,1,423,
%T A182906 370,171,44,5,978,882,437,129,20,1,2283,2112,1107,364,70,6,5373,5079,
%U A182906 2790,1000,225,27,1,12735,12264
%N A182906 Triangle read by rows: T(n,k) is the number of weighted lattice paths in F[n] having endpoint height k (k<=floor(n/2)). The members of F[n] are paths of weight n that start at (0,0), do not go below 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, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
%C A182906 The paths need not end on the horizontal axis.
%C A182906 Number of entries in row n is 1+floor(n/2).
%C A182906 Sum of entries in row n is A182905(n).
%C A182906 T(n,0) = A004148(n+1).
%C A182906 T(n,1) = A166297(n+1).
%H A182906 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 A182906 G.f.: G(t,z) = g/(1-tz^2*g), where g=g(z) is defined by g = 1 + z*g + z^2*g + z^3*g^2.
%F A182906 Rec. rel.: T(n,k) = T(n-1,k) + T(n-1,k+1) + T(n-2,k) + T(n-2,k-1); the 2nd Maple program makes use of this.
%F A182906 T(n,k) = (k+1)*Sum_{i=0..n-k+1} C(i+1,n+1-i)*C(i+1,-i+n-k)/(i+1). - _Vladimir Kruchinin_, Jan 25 2019
%e A182906 T(4,1)=5. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hhU, hUh, Uhh, HU, and UH.
%e A182906 Triangle starts:
%e A182906 1;
%e A182906 1;
%e A182906 2, 1;
%e A182906 4, 2;
%e A182906 8, 5, 1;
%e A182906 17, 12, 3;
%e A182906 37, 28, 9, 1;
%p A182906 g := ((1-z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))*1/2)/z^3: G := g/(1-t*z^2*g); Gser := simplify(series(G, z = 0, 22)): for n from 0 to 14 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%p A182906 T := proc (n, k) if k < 0 then 0 elif n < 0 then 0 elif (1/2)*n < k then 0 elif n = 0 and k = 0 then 1 else T(n-1, k)+T(n-1, 1+k)+T(n-2, k)+T(n-2, k-1) end if end proc: for n from 0 to 14 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do;
%o A182906 (Maxima)
%o A182906 T(n,k):=(k+1)*sum((binomial(i+1,n+1-i)*binomial(i+1,-i+n-k))/(i+1),i,0,n-k+1);
%o A182906 /* _Vladimir Kruchinin_, Jan 25 2019 */
%Y A182906 Cf. A182905, A004148, A166297.
%K A182906 nonn,tabf
%O A182906 0,3
%A A182906 _Emeric Deutsch_, Dec 17 2010