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A296664 Table read by rows, diagonals of powers of Toeplitz matrices generated by the characteristic function of 1, T(n, k) for n >= 0 and 0 <= k <= 2*floor(n/2).

%I A296664 #25 Mar 08 2020 05:00:21
%S A296664 1,1,1,2,1,2,3,2,2,5,6,5,2,5,9,10,9,5,5,14,19,20,19,14,5,14,28,34,35,
%T A296664 34,28,14,14,42,62,69,70,69,62,42,14,42,90,117,125,126,125,117,90,42,
%U A296664 42,132,207,242,251,252,251,242,207,132,42
%N A296664 Table read by rows, diagonals of powers of Toeplitz matrices generated by the characteristic function of 1, T(n, k) for n >= 0 and 0 <= k <= 2*floor(n/2).
%C A296664 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(n)^n if n is even or the diagonal next to the main diagonal if n is odd. Note that the antidiagonals of M(n)^n are the rows of Pascal's triangle A007318.
%F A296664 T(n, 0) = T(n, 2*floor(n/2)) = A208355(n) = A000108(floor((n+1)/2)).
%F A296664 T(n, floor(n/2)) = A001405(n).
%F A296664 Further formulas can be found in A296662 and A296666 for the cases n odd and n even.
%e A296664 The first few matrices M(n)^n are:
%e A296664 n=0   n=1     n=2       n=3         n=4
%e A296664 |1|  |0 1|  |1 0 1|  |0 2 0 1|  |2 0 3 0 1|
%e A296664      |1 0|  |0 2 0|  |2 0 3 0|  |0 5 0 4 0|
%e A296664             |1 0 1|  |0 3 0 2|  |3 0 6 0 3|
%e A296664                      |1 0 2 0|  |0 4 0 5 0|
%e A296664                                 |1 0 3 0 2|
%e A296664 The triangle starts:
%e A296664 0: [ 1]
%e A296664 1: [ 1]
%e A296664 2: [ 1,  2,   1]
%e A296664 3: [ 2,  3,   2]
%e A296664 4: [ 2,  5,   6,   5,   2]
%e A296664 5: [ 5,  9,  10,   9,   5]
%e A296664 6: [ 5, 14,  19,  20,  19,  14,  5]
%e A296664 7: [14, 28,  34,  35,  34,  28,  14]
%e A296664 8: [14, 42,  62,  69,  70,  69,  62, 42, 14]
%e A296664 9: [42, 90, 117, 125, 126, 125, 117, 90, 42]
%p A296664 v := n -> `if`(n=1, 1, 0):
%p A296664 M := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j), j=0..n)], symmetric):
%p A296664 seq(convert(ArrayTools:-Diagonal(M(n)^n, n mod 2), list), n=0..10);
%t A296664 v[n_] := If[n == 1, 1, 0];
%t A296664 m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
%t A296664 d[n_] := If[n == 0, {1}, Diagonal[m[n], Mod[n, 2]]];
%t A296664 Table[d[n], {n, 0, 10}] // Flatten
%o A296664 (Sage)
%o A296664 def T(n, k):
%o A296664     h, e = n//2, n%2 == 0
%o A296664     a = binomial(n, h) if e else binomial(2*h+1, h+1)
%o A296664     if k > h:
%o A296664         b = binomial(n, k-h-1) if e else binomial(2*h+1, k-h-1)
%o A296664     else:
%o A296664         b = binomial(n, h+k+1) if e else binomial(2*h+1, h-k-1)
%o A296664     return a - b
%o A296664 for n in (0..9): print([T(n, k) for k in (0..2*(n//2))])
%Y A296664 Cf. A000108, A001405, A208355, A296663 (row sums), A296662 (odd rows), A296666 (even rows).
%K A296664 nonn,tabf
%O A296664 0,4
%A A296664 _Peter Luschny_, Dec 19 2017