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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.

%I A322790 #52 Dec 29 2018 14:32:05
%S A322790 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,
%T A322790 161,11,1,1,19601,47525,18817,2889,241,13,1,1,114243,470449,262087,
%U A322790 51841,5291,337,15,1,1,665857,4656965,3650401,930249,116161,8749,449,17,1
%N 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.
%H A322790 Seiichi Manyama, <a href="/A322790/b322790.txt">Antidiagonals n = 0..139, flattened</a>
%H A322790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H A322790 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A322790 a(n) = 2 * A322699(n) + 1.
%F A322790 A(n,k) + sqrt(A(n,k)^2 - 1) = (sqrt(n+1) + sqrt(n))^(2*k).
%F A322790 A(n,k) - sqrt(A(n,k)^2 - 1) = (sqrt(n+1) - sqrt(n))^(2*k).
%F A322790 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.
%F A322790 A(n,k) = T_{k}(2*n+1) where T_{k}(x) is a Chebyshev polynomial of the first kind.
%F A322790 T_1(x) = x. So A(n,1) = 2*n+1.
%e A322790 Square array begins:
%e A322790    1,  1,   1,    1,      1,       1,         1, ...
%e A322790    1,  3,  17,   99,    577,    3363,     19601, ...
%e A322790    1,  5,  49,  485,   4801,   47525,    470449, ...
%e A322790    1,  7,  97, 1351,  18817,  262087,   3650401, ...
%e A322790    1,  9, 161, 2889,  51841,  930249,  16692641, ...
%e A322790    1, 11, 241, 5291, 116161, 2550251,  55989361, ...
%e A322790    1, 13, 337, 8749, 227137, 5896813, 153090001, ...
%t A322790 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 *)
%Y A322790 Columns 0-3 give A000012, A005408, A069129(n+1), A322830.
%Y A322790 Rows 0-9 give A000012, A001541, A001079, A011943(n+1), A023039, A077422, A097308, A068203, A056771, A078986.
%Y A322790 Main diagonal gives A173174.
%Y A322790 A(n-1,n) gives A173148(n).
%Y A322790 Cf. A322699, A322747.
%K A322790 nonn,tabl
%O A322790 0,5
%A A322790 _Seiichi Manyama_, Dec 26 2018