A322790 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j.
1, 1, 1, 1, 3, 1, 1, 17, 5, 1, 1, 99, 49, 7, 1, 1, 577, 485, 97, 9, 1, 1, 3363, 4801, 1351, 161, 11, 1, 1, 19601, 47525, 18817, 2889, 241, 13, 1, 1, 114243, 470449, 262087, 51841, 5291, 337, 15, 1, 1, 665857, 4656965, 3650401, 930249, 116161, 8749, 449, 17, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 3, 17, 99, 577, 3363, 19601, ... 1, 5, 49, 485, 4801, 47525, 470449, ... 1, 7, 97, 1351, 18817, 262087, 3650401, ... 1, 9, 161, 2889, 51841, 930249, 16692641, ... 1, 11, 241, 5291, 116161, 2550251, 55989361, ... 1, 13, 337, 8749, 227137, 5896813, 153090001, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
- Wikipedia, Chebyshev polynomials.
- Index entries for sequences related to Chebyshev polynomials.
Crossrefs
Programs
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Mathematica
A[0, k_] := 1; A[n_, k_] := Sum[Binomial[2 k, 2 j]*(n + 1)^(k - j)*n^j, {j, 0, k}]; Table[A[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 26 2018 *)
Formula
a(n) = 2 * A322699(n) + 1.
A(n,k) + sqrt(A(n,k)^2 - 1) = (sqrt(n+1) + sqrt(n))^(2*k).
A(n,k) - sqrt(A(n,k)^2 - 1) = (sqrt(n+1) - sqrt(n))^(2*k).
A(n,0) = 1, A(n,1) = 2*n+1 and A(n,k) = (4*n+2) * A(n,k-1) - A(n,k-2) for k > 1.
A(n,k) = T_{k}(2*n+1) where T_{k}(x) is a Chebyshev polynomial of the first kind.
T_1(x) = x. So A(n,1) = 2*n+1.