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 A370262 #17 Apr 17 2024 03:42:55
%S A370262 1,1,1,1,5,5,1,14,49,49,1,30,243,729,729,1,55,847,5324,14641,14641,1,
%T A370262 91,2366,26364,142805,371293,371293,1,140,5670,101250,928125,4556250,
%U A370262 11390625,11390625,1,204,12138,324258,4593655,36916282,168962983,410338673,410338673
%N A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.
%C A370262 The table entries are integers since a(n, k) := binomial(n+k, n-k)/(2*k + 1) * (2*n + 1) gives the entries of the transpose of triangle A082985.
%F A370262 n-th row polynomial R(n, x) = Sum_{k = 0..n} T(n, k)*x^k = sqrt( 2* Sum_{k = 0..2*n} (2*n + 1)^(k-1) *binomial(2*n+k+2, 2*k+2)/(2*n + k + 2) * x^k ).
%F A370262 R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
%F A370262 R(n, 2) = A370260(n).
%e A370262 Triangle begins
%e A370262 n\k | 0 1 2 3 4 5 6
%e A370262 - - - - - - - - - - - - - - - - - - - - - - - - - - - -
%e A370262 0 | 1
%e A370262 1 | 1 1
%e A370262 2 | 1 5 5
%e A370262 3 | 1 14 49 49
%e A370262 4 | 1 30 243 729 729
%e A370262 5 | 1 55 847 5324 14641 14641
%e A370262 6 | 1 91 2366 26364 142805 371293 371293
%e A370262 ...
%p A370262 seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);
%t A370262 Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Apr 17 2024 *)
%Y A370262 A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).
%Y A370262 Cf. A008310, A082985, A258708, A370259, A370260.
%K A370262 nonn,tabl,easy
%O A370262 0,5
%A A370262 _Peter Bala_, Mar 12 2024