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A123634 Upper half of Hankel determinant number wall for A004148.

%I A123634 #6 Dec 31 2016 11:52:55
%S A123634 1,1,1,1,1,1,1,2,0,0,1,4,0,0,-1,1,8,4,-2,-1,-1,1,17,7,3,-3,-1,-1,1,37,
%T A123634 25,6,-6,-4,0,0,1,82,121,-38,-4,-16,0,0,1,1,185,461,160,-104,-64,-16,
%U A123634 4,1,1,1,423,2001,588,-144,-360,-60,-10,5,1,1,1,978,9225,360,1836,-2160,-450,-50,15,6,0,0,1
%N A123634 Upper half of Hankel determinant number wall for A004148.
%F A123634 T(n, 0) = 1. T(n, 1) = a(n) if n>0, T(n, 2) = a(n+1)*a(n-1) - a(n)^2 if n>1, T(n, 3) = det([a(n-2), a(n-1), a(n); a(n-1), a(n), a(n+1); a(n), a(n+1), a(n+2)]) if n>2 where a(n) = A004148(n).
%F A123634 T(n, n) = A046978(n+1). T(n+1, n) = A132380(n+2). - _Michael Somos_, Dec 31 2016
%e A123634 Table is:
%e A123634 n\k  0   1   2   3   4   5   6
%e A123634 --  --  --  --  --  --  --  --
%e A123634 0 |  1
%e A123634 1 |  1   1
%e A123634 2 |  1   1   1
%e A123634 3 |  1   2   0   0
%e A123634 4 |  1   4   0   0  -1
%e A123634 5 |  1   8   4  -2  -1  -1
%e A123634 6 |  1  17   7   3  -3  -1  -1
%o A123634 (PARI) {T(n, k) = my(m); if( k<0 || k>n, 0, matdet( matrix(k, k, i, j, polcoeff( (1 - x + x^2 - sqrt(1 - 2*x - x^2 + x^3*(-2 + x + O(x^(m=i+j+n-k-1))))) / (2*x^2), m))))};
%Y A123634 Cf. A004148, A046978, A132380.
%K A123634 sign,tabl
%O A123634 0,8
%A A123634 _Michael Somos_, Oct 04 2006