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A322699 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is 1/2 * (-1 + Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j). 9

%I

%S 0,0,0,0,1,0,0,8,2,0,0,49,24,3,0,0,288,242,48,4,0,0,1681,2400,675,80,

%T 5,0,0,9800,23762,9408,1444,120,6,0,0,57121,235224,131043,25920,2645,

%U 168,7,0,0,332928,2328482,1825200,465124,58080,4374,224,8,0

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is 1/2 * (-1 + Sum_{j=0..k} binomial(2*k,2*j)*(n+1)^(k-j)*n^j).

%H Seiichi Manyama, <a href="/A322699/b322699.txt">Antidiagonals n = 0..139, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F sqrt(A(n,k)+1) + sqrt(A(n,k)) = (sqrt(n+1) + sqrt(n))^k.

%F sqrt(A(n,k)+1) - sqrt(A(n,k)) = (sqrt(n+1) - sqrt(n))^k.

%F A(n,0) = 0, A(n,1) = n and A(n,k) = (4*n+2) * A(n,k-1) - A(n,k-2) + 2*n for k > 1.

%F A(n,k) = (T_{k}(2*n+1) - 1)/2 where T_{k}(x) is a Chebyshev polynomial of the first kind.

%F T_1(x) = x. So A(n,1) = (2*n+1-1)/2 = n.

%e Square array begins:

%e 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 8, 49, 288, 1681, 9800, ...

%e 0, 2, 24, 242, 2400, 23762, 235224, ...

%e 0, 3, 48, 675, 9408, 131043, 1825200, ...

%e 0, 4, 80, 1444, 25920, 465124, 8346320, ...

%e 0, 5, 120, 2645, 58080, 1275125, 27994680, ...

%e 0, 6, 168, 4374, 113568, 2948406, 76545000, ...

%t Unprotect[Power]; 0^0 := 1; Protect[Power]; Table[(-1 + Sum[Binomial[2 k, 2 j] (# + 1)^(k - j)*#^j, {j, 0, k}])/2 &[n - k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* _Michael De Vlieger_, Jan 01 2019 *)

%t nmax = 9; row[n_] := LinearRecurrence[{4n+3, -4n-3, 1}, {0, n, 4n(n+1)}, nmax+1]; T = Array[row, nmax+1, 0]; A[n_, k_] := T[[n+1, k+1]];

%t Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* _Jean-Fran├žois Alcover_, Jan 06 2019 *)

%o (Ruby)

%o def ncr(n, r)

%o return 1 if r == 0

%o (n - r + 1..n).inject(:*) / (1..r).inject(:*)

%o end

%o def A(k, n)

%o (0..n).map{|i| (0..k).inject(-1){|s, j| s + ncr(2 * k, 2 * j) * (i + 1) ** (k - j) * i ** j} / 2}

%o end

%o def A322699(n)

%o a = []

%o (0..n).each{|i| a << A(i, n - i)}

%o ary = []

%o (0..n).each{|i|

%o (0..i).each{|j|

%o ary << a[i - j][j]

%o }

%o }

%o ary

%o end

%o p A322699(10)

%Y Columns 0-5 give A000004, A001477, A033996, A322675, A322677, A322745.

%Y Rows 0-9 give A000004, A001108, A132596, A007654(n+1), A132584, A322707, A322708, A322709, A132592, A132593.

%Y Main diagonal gives A322746.

%Y Cf. A173175 (A(n,2*n)), A322790.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Dec 23 2018

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Last modified March 31 02:36 EDT 2020. Contains 333135 sequences. (Running on oeis4.)