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A123634
Upper half of Hankel determinant number wall for A004148.
0
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
0,8
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
EXAMPLE
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
PROG
(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))))};
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Michael Somos, Oct 04 2006
STATUS
approved