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 A272866 #23 Jan 28 2025 10:51:54
%S A272866 1,1,3,1,1,6,11,6,1,1,9,30,45,30,9,1,1,12,58,144,195,144,58,12,1,1,15,
%T A272866 95,330,685,873,685,330,95,15,1,1,18,141,630,1770,3258,3989,3258,1770,
%U A272866 630,141,18,1,1,21,196,1071,3801,9198,15533,18483,15533,9198,3801,1071,196,21,1
%N A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
%C A272866 From _R. J. Mathar_, Nov 05 2021: (Start)
%C A272866 These are the antidiagonals of the following array with the bivariate generating function 1/(1-x^2-3*x*y-y^2):
%C A272866 1 0 1 0 1 0 1 0 1 0 1 ...
%C A272866 0 3 0 6 0 9 0 12 0 15 0 ...
%C A272866 1 0 11 0 30 0 58 0 95 0 141 ...
%C A272866 0 6 0 45 0 144 0 330 0 630 0 ...
%C A272866 1 0 30 0 195 0 685 0 1770 0 3801 ...
%C A272866 0 9 0 144 0 873 0 3258 0 9198 0 ...
%C A272866 1 0 58 0 685 0 3989 0 15533 0 46928 ...
%C A272866 0 12 0 330 0 3258 0 18483 0 74280 0 ...
%C A272866 1 0 95 0 1770 0 15533 0 86515 0 356283 ...
%C A272866 0 15 0 630 0 9198 0 74280 0 408105 0 ...
%C A272866 1 0 141 0 3801 0 46928 0 356283 0 1936881 ... (End)
%H A272866 Michael De Vlieger, <a href="/A272866/b272866.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 100, flattened).
%H A272866 Feryal Alayont and Evan Henning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Alayont/ala4.html">Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
%H A272866 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 6.
%H A272866 László Németh, <a href="http://math.colgate.edu/~integers/t41/t41.Abstract.html">Tetrahedron trinomial coefficient transform</a>, Integers (2019) 19, Article A41.
%F A272866 T(n,n) = A026375(n) for n>=0.
%F A272866 T(n,n-1) = A026376(n) for n>=1.
%F A272866 T(n,n+1)/n = A002212(n) for n>=1.
%e A272866 1;
%e A272866 1, 3, 1;
%e A272866 1, 6, 11, 6, 1;
%e A272866 1, 9, 30, 45, 30, 9, 1;
%e A272866 1, 12, 58, 144, 195, 144, 58, 12, 1;
%e A272866 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1;
%p A272866 T := (n,k) -> simplify(GegenbauerC(`if`(k<n,k,2*n-k),-n, -3/2)):
%p A272866 for n from 0 to 6 do seq(T(n,k),k=0..2*n) od;
%t A272866 Table[If[n == 0, 1, GegenbauerC[If[k < n, k, 2 n - k], -n, -3/2]], {n, 0, 7}, {k, 0, 2 n}] // Flatten (* _Michael De Vlieger_, Aug 02 2019 *)
%Y A272866 Cf. A002212, A026375, A026376, A110165.
%K A272866 nonn,tabf
%O A272866 0,3
%A A272866 _Peter Luschny_, May 08 2016