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 A324242 #58 Sep 03 2019 15:20:41
%S A324242 1,1,3,1,4,1,5,7,1,6,11,1,7,16,18,1,8,22,29,1,9,29,45,47,1,10,37,67,
%T A324242 76,1,11,46,96,121,123,1,12,56,133,188,199,1,13,67,179,284,320,322,1,
%U A324242 14,79,235,417,508,521,1,15,92,302,596,792,841,843,1,16,106,381,831,1209,1349,1364
%N A324242 Incomplete Lucas numbers: irregular triangular array L(n,k) = Sum_{j = 0..k} (n/(n-j)) * binomial(n-j, j), read by rows, with n >= 1 and 0 <= k <= floor(n/2).
%C A324242 For additional properties of the incomplete Lucas numbers and special cases not listed here, see Filipponi (1996, pp. 45-53).
%H A324242 A. Dil and I. Mezo, <a href="http://dx.doi.org/10.1016/j.amc.2008.10.013">A symmetric algorithm for hyperharmonic and Fibonacci numbers</a>, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Lucas numbers.
%H A324242 Piero Filipponi, <a href="https://doi.org/10.1007/BF02845088">Incomplete Fibonacci and Lucas numbers</a>, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 2 (p. 46) that contains the incomplete Lucas numbers.
%H A324242 A. Pintér and H.M. Srivastava, <a href="https://doi.org/10.1007/BF02844348">Generating functions of the incomplete Fibonacci and Lucas numbers</a>, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.
%F A324242 L(n,k) = F(n-1, k-1) + F(n+1, k) for n >= 1 and 0 <= k <= floor(n/2), where F(n,k) = Sum_{j = 0..k} binomial(n-1-j, j) are the incomplete Fibonacci numbers (defined for n >= 1 and 0 <= k <= floor((n-1)/2)).
%F A324242 L(n+2, k+1) = L(n+1, k+1) + L(n,k) for n >= 1 and 0 <= k <= floor((n-1)/2).
%F A324242 L(n,k) = F(n+2,k) - F(n-2, k-2) for n >= 3 and 2 <= k <= floor((n+1)/2).
%F A324242 Special cases: L(n,0) = 1 (n >= 1), L(n,1) = n+1 (n >= 2), L(n,2) = (n^2-n+2)/2 = A000124(n-1) (n >= 4), and L(n, floor(n/2)) = A000204(n) (n >= 1).
%F A324242 Sum of row n = (3 + (-1)^n)*A000204(n)/4 + n*A000045(n)/2.
%F A324242 G.f. for column k >= 1: t^(2*k)*((A000204(2*k) + t*A000204(2*k-1))*(1-t)^(k+1) - t^2*(2-t))/((1-t)^(k+1) * (1-t-t^2)).
%e A324242 Triangle L(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
%e A324242 1;
%e A324242 1, 3;
%e A324242 1, 4;
%e A324242 1, 5, 7;
%e A324242 1, 6, 11;
%e A324242 1, 7, 16, 18;
%e A324242 1, 8, 22, 29;
%e A324242 1, 9, 29, 45, 47;
%e A324242 1, 10, 37, 67, 76;
%e A324242 1, 11, 46, 96, 121, 123;
%e A324242 1, 12, 56, 133, 188, 199;
%e A324242 ...
%e A324242 Row sums are 1, 4, 5, 13, 18, 42, 60, 131, 191, 398, 589, 1186, 1775, 3482, 5257, 10103, 15360, ...
%t A324242 Flatten[Table[Sum[(n/(n-j))*Binomial[n-j, j],{j,0,k}],{n,1,15},{k,0,Floor[n/2]}]] (* _Stefano Spezia_, Sep 03 2019 *)
%Y A324242 Cf. A038730, A038792, and A134511 for various versions of the incomplete Fibonacci numbers.
%Y A324242 Cf. A000032, A000045, A000124, A000204, A137176.
%K A324242 nonn,tabf
%O A324242 1,3
%A A324242 _Petros Hadjicostas_, Sep 02 2019