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A296662 Table read by rows, the odd rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.

%I A296662 #16 Mar 08 2020 04:58:25
%S A296662 1,2,3,2,5,9,10,9,5,14,28,34,35,34,28,14,42,90,117,125,126,125,117,90,
%T A296662 42,132,297,407,451,461,462,461,451,407,297,132,429,1001,1430,1638,
%U A296662 1703,1715,1716,1715,1703,1638,1430,1001,429
%N A296662 Table read by rows, the odd rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.
%C A296662 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 diagonal next to the main diagonal of M(2n+1)^(2n+1).
%F A296662 T(n, n) = A001700(n).
%F A296662 T(n, 0) = T(n, 2*n) = A000108(n+1).
%F A296662 T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1) for k=0..n.
%F A296662 T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, k-n-1) for k=n+1..2*n and n>0.
%e A296662 The triangle starts:
%e A296662 0: [  1]
%e A296662 1: [  2,   3,   2]
%e A296662 2: [  5,   9,  10,   9,   5]
%e A296662 3: [ 14,  28,  34,  35,  34,  28,  14]
%e A296662 4: [ 42,  90, 117, 125, 126, 125, 117,  90,  42]
%e A296662 5: [132, 297, 407, 451, 461, 462, 461, 451, 407, 297, 132]
%p A296662 v := n -> `if`(n=1, 1, 0):
%p A296662 B := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j),j=0..n)], symmetric):
%p A296662 seq(convert(ArrayTools:-Diagonal(B(2*n+1)^(2*n+1), 1),list), n=0..6);
%t A296662 v[n_] := If[n == 1, 1, 0];
%t A296662 m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
%t A296662 d[n_] := Diagonal[m[2 n + 1], 1];
%t A296662 Table[d[n], {n, 0, 6}] // Flatten
%o A296662 (Sage)
%o A296662 def T(n, k):
%o A296662     if k > n:
%o A296662         b = binomial(2*n+1, k - n - 1)
%o A296662     else:
%o A296662         b = binomial(2*n+1, n - k - 1)
%o A296662     return binomial(2*n+1, n+1) - b
%o A296662 for n in (0..6):
%o A296662     print([T(n, k) for k in (0..2*n)])
%Y A296662 Cf. A000108, A001700, A050158, A296664, A296666, A296769 (row sums).
%K A296662 nonn,tabf
%O A296662 0,2
%A A296662 _Peter Luschny_, Dec 20 2017