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 A342133 #36 Dec 12 2023 20:22:01
%S A342133 1,1,0,1,2,0,1,4,3,0,1,6,14,4,0,1,8,33,48,5,0,1,10,60,180,164,6,0,1,
%T A342133 12,95,448,981,560,7,0,1,14,138,900,3344,5346,1912,8,0,1,16,189,1584,
%U A342133 8525,24960,29133,6528,9,0,1,18,248,2548,18180,80750,186304,158760,22288,10,0
%N A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x + k*x^2).
%H A342133 Seiichi Manyama, <a href="/A342133/b342133.txt">Antidiagonals n = 0..139, flattened</a>
%H A342133 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A342133 T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.
%F A342133 T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).
%F A342133 T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.
%e A342133 Square array begins:
%e A342133 1, 1, 1, 1, 1, 1, ...
%e A342133 0, 2, 4, 6, 8, 10, ...
%e A342133 0, 3, 14, 33, 60, 95, ...
%e A342133 0, 4, 48, 180, 448, 900, ...
%e A342133 0, 5, 164, 981, 3344, 8525, ...
%e A342133 0, 6, 560, 5346, 24960, 80750, ...
%p A342133 T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
%p A342133 seq(seq(T(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Mar 01 2021
%t A342133 T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Apr 27 2021 *)
%o A342133 (PARI) T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));
%o A342133 (PARI) T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));
%o A342133 (PARI) T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));
%Y A342133 Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).
%Y A342133 Rows 0..2 give A000012, A005843, A033991.
%Y A342133 Main diagonal gives (-1) * A109520(n+1).
%Y A342133 Cf. A342120, A342129, A342134.
%K A342133 nonn,tabl
%O A342133 0,5
%A A342133 _Seiichi Manyama_, Mar 01 2021