A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k.
1, 1, 1, 1, 5, 5, 1, 14, 49, 49, 1, 30, 243, 729, 729, 1, 55, 847, 5324, 14641, 14641, 1, 91, 2366, 26364, 142805, 371293, 371293, 1, 140, 5670, 101250, 928125, 4556250, 11390625, 11390625, 1, 204, 12138, 324258, 4593655, 36916282, 168962983, 410338673, 410338673
Offset: 0
Examples
Triangle begins n\k | 0 1 2 3 4 5 6 - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 | 1 1 | 1 1 2 | 1 5 5 3 | 1 14 49 49 4 | 1 30 243 729 729 5 | 1 55 847 5324 14641 14641 6 | 1 91 2366 26364 142805 371293 371293 ...
Crossrefs
Programs
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Maple
seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);
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Mathematica
Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)
Formula
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 ).
R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
R(n, 2) = A370260(n).
Comments