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 A342134 #32 Apr 27 2021 02:11:43
%S A342134 1,1,0,1,2,0,1,4,5,0,1,6,18,12,0,1,8,39,80,29,0,1,10,68,252,356,70,0,
%T A342134 1,12,105,576,1629,1584,169,0,1,14,150,1100,4880,10530,7048,408,0,1,
%U A342134 16,203,1872,11525,41344,68067,31360,985,0,1,18,264,2940,23364,120750,350272,439992,139536,2378,0
%N A342134 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 A342134 Seiichi Manyama, <a href="/A342134/b342134.txt">Antidiagonals n = 0..139, flattened</a>
%H A342134 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A342134 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 A342134 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 A342134 T(n,k) = (-sqrt(k)*i)^n * U(n, sqrt(k)*i) where U(n, x) is a Chebyshev polynomial of the second kind.
%e A342134 Square array begins:
%e A342134 1, 1, 1, 1, 1, 1, ...
%e A342134 0, 2, 4, 6, 8, 10, ...
%e A342134 0, 5, 18, 39, 68, 105, ...
%e A342134 0, 12, 80, 252, 576, 1100, ...
%e A342134 0, 29, 356, 1629, 4880, 11525, ...
%e A342134 0, 70, 1584, 10530, 41344, 120750, ...
%p A342134 T:= (n, k)-> (<<0|1>, <k|2*k>>^(n+1))[1, 2]:
%p A342134 seq(seq(T(n, d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Mar 01 2021
%t A342134 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 A342134 (PARI) T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*2^(-j)*binomial(n-j, j));
%o A342134 (PARI) T(n, k) = sum(j=0, n, (2*k)^j*2^(j-n)*binomial(j, n-j));
%o A342134 (PARI) T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I));
%Y A342134 Columns 0..5 give A000007, A000129(n+1), A090017(n+1), A090018, A190510(n+1), A190955(n+1).
%Y A342134 Rows 0..2 give A000012, A005843, A007742.
%Y A342134 Main diagonal gives A109517(n+1).
%Y A342134 Cf. A342120, A342129, A342133.
%K A342134 nonn,tabl
%O A342134 0,5
%A A342134 _Seiichi Manyama_, Mar 01 2021