A370262 - OEIS

%I #17 Apr 17 2024 03:42:55

%S 1,1,1,1,5,5,1,14,49,49,1,30,243,729,729,1,55,847,5324,14641,14641,1,

%T 91,2366,26364,142805,371293,371293,1,140,5670,101250,928125,4556250,

%U 11390625,11390625,1,204,12138,324258,4593655,36916282,168962983,410338673,410338673

%N Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.

%C 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 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 R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).

%F R(n, 2) = A370260(n).

%e Triangle begins

%e  n\k | 0    1      2       3        4        5        6

%e  - - - - - - - - - - - - - - - - - - - - - - - - - - - -

%e    0 | 1

%e    1 | 1    1

%e    2 | 1    5      5

%e    3 | 1   14     49      49

%e    4 | 1   30    243     729      729

%e    5 | 1   55    847    5324    14641    14641

%e    6 | 1   91   2366   26364   142805   371293   371293

%e   ...

%p seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);

%t 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 A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).

%Y Cf. A008310, A082985, A258708, A370259, A370260.

%K nonn,tabl,easy

%O 0,5

%A _Peter Bala_, Mar 12 2024