A094384 Determinant of n X n partial Hadamard matrix with coefficient m(i,j) 1<=i,j<=n (see comment).

1, -2, 4, 16, -32, -128, -512, 4096, -8192, -32768, -131072, 1048576, 4194304, -33554432, 268435456, 4294967296, -8589934592, -34359738368, -137438953472, 1099511627776, 4398046511104, -35184372088832, 281474976710656

Offset: 1

Author

Benoit Cloitre, Jun 03 2004

Keywords

• sign

Comments

Let M(infinity) be the infinite matrix with coefficient m(i,j) i>=1, j>=1 defined as follows : M(0)=1 and M(k) is the 2^k X 2^k matrix following the recursion : +M(k-1)-M(k-1) M(k)= -M(k-1)-M(k-1)

Examples

			M(2)=/1,-1/-1,-1/ then a(2)=detM(2)=-2
		

Crossrefs

Cf. A000788, A073536.

Programs

Python

from sympy import Matrix
def A094384(n):
    m = Matrix([1])
    for i in range((n-1).bit_length()):
        m = Matrix([[m, -m],[-m, -m]])
    return m[:n,:n].det() # Chai Wah Wu, Nov 12 2024

Formula

It appears that abs(a(n))=2^A000788(n). What is the rule for signs? Does sum(k=1, n, a(k+1)/a(k))=0 iff n is in A073536 ?

Conjecture: a(n) = (-2)^A000788(n-1). - Chai Wah Wu, Nov 12 2024