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 A257365 #20 Jan 21 2019 12:00:45
%S A257365 1,1,1,1,2,1,1,4,3,1,1,6,8,4,1,1,8,16,13,5,1,1,10,28,32,19,6,1,1,12,
%T A257365 44,68,55,26,7,1,1,14,64,128,136,86,34,8,1,1,16,88,220,296,241,126,43,
%U A257365 9,1,1,18,116,352,584,592,393,176,53,10,1
%N A257365 Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)*C(n-2*m,k).
%C A257365 From _Emanuele Munarini_, Feb 21 2017: (Start)
%C A257365 T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps X=(1,0), D=(1,1) and E=(3,1).
%C A257365 Row sums = A008998.
%C A257365 Central coefficients = A006139. (End)
%H A257365 Michael De Vlieger, <a href="/A257365/b257365.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A257365 James East, Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%F A257365 G.f.: 1/(1-y-x*(1+y^2)).
%F A257365 From _Emanuele Munarini_, Feb 21 2017: (Start)
%F A257365 G.f. for the triangle: 1/(1-x-x*y-x^3*y).
%F A257365 Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n,k). (End)
%e A257365 1;
%e A257365 1, 1;
%e A257365 1, 2, 1;
%e A257365 1, 4, 3, 1;
%e A257365 1, 6, 8, 4, 1;
%e A257365 1, 8, 16, 13, 5, 1;
%t A257365 Table[Sum[Binomial[k, m] Binomial[n - 2 m, k], {m, 0, (n - k)/2}], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Apr 21 2015 *)
%o A257365 (Maxima) T(n,k):=sum(binomial(k,m)*binomial(n-2*m,k),m,0,(n-k)/2);
%Y A257365 Cf. A006139.
%K A257365 nonn,tabl
%O A257365 0,5
%A A257365 _Vladimir Kruchinin_, Apr 21 2015