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A296666 Table read by rows, the even rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.

%I A296666 #33 Mar 07 2020 14:57:22
%S A296666 1,1,2,1,2,5,6,5,2,5,14,19,20,19,14,5,14,42,62,69,70,69,62,42,14,42,
%T A296666 132,207,242,251,252,251,242,207,132,42,132,429,704,858,912,923,924,
%U A296666 923,912,858,704,429,132
%N A296666 Table read by rows, the even rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.
%C A296666 Let v be the characteristic function of 1 (A063524) and M(n) for n >= 0 the symmetric Toeplitz matrix generated by the initial segment of v, then row n is the main diagonal of M(2n)^(2n).
%C A296666 Seems to be A050157 + its reflection. - _Andrey Zabolotskiy_, Dec 19 2017
%H A296666 Peter Luschny, <a href="/A296666/b296666.txt">Row n for n = 0..30</a>
%F A296666 T(n, 0) = T(n, 2*n) = A000108(n).
%F A296666 T(n, n) are the central binomial coefficients A000984(n).
%F A296666 T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1) for k=0..n.
%F A296666 T(n, k) = binomial(2*n, n) - binomial(2*n, k-n-1) for k=n+1..2*n and n>0.
%e A296666 0: [  1]
%e A296666 1: [  1,   2,   1]
%e A296666 2: [  2,   5,   6,   5,   2]
%e A296666 3: [  5,  14,  19,  20,  19,  14,   5]
%e A296666 4: [ 14,  42,  62,  69,  70,  69,  62,  42,  14]
%e A296666 5: [ 42, 132, 207, 242, 251, 252, 251, 242, 207, 132,  42]
%e A296666 6: [132, 429, 704, 858, 912, 923, 924, 923, 912, 858, 704, 429, 132]
%p A296666 v := n -> `if`(n=1, 1, 0);
%p A296666 B := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j), j=0..n)], symmetric):
%p A296666 seq(convert(ArrayTools:-Diagonal(B(2*n)^(2*n)), list), n=0..10);
%t A296666 v[n_] := If[n == 1, 1, 0];
%t A296666 m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
%t A296666 d[n_] := If[n == 0, {1}, Diagonal[m[2 n]]];
%t A296666 Table[d[n], {n, 0, 6}] // Flatten
%o A296666 (Sage)
%o A296666 def T(n, k):
%o A296666     if k > n:
%o A296666         b = binomial(2*n, k - n - 1)
%o A296666     else:
%o A296666         b = binomial(2*n, n + k + 1)
%o A296666     return binomial(2*n, n) - b
%o A296666 for n in (0..6):
%o A296666     print([T(n, k) for k in (0..2*n)])
%Y A296666 Cf. A000108, A000984, A050157, A296662, A296664, A296665 (row sums).
%K A296666 nonn,tabf
%O A296666 0,3
%A A296666 _Peter Luschny_, Dec 19 2017