A123634 Upper half of Hankel determinant number wall for A004148.

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, 25, 6, -6, -4, 0, 0, 1, 82, 121, -38, -4, -16, 0, 0, 1, 1, 185, 461, 160, -104, -64, -16, 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

Offset: 0

Author

Michael Somos, Oct 04 2006

Keywords

• sign
• tabl

Examples

			Table is:
n\k  0   1   2   3   4   5   6
--  --  --  --  --  --  --  --
0 |  1
1 |  1   1
2 |  1   1   1
3 |  1   2   0   0
4 |  1   4   0   0  -1
5 |  1   8   4  -2  -1  -1
6 |  1  17   7   3  -3  -1  -1
		

Crossrefs

Cf. A004148, A046978, A132380.

Programs

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))))};

Formula

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).

T(n, n) = A046978(n+1). T(n+1, n) = A132380(n+2). - Michael Somos, Dec 31 2016