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

%I A094384 #10 Nov 13 2024 17:16:11
%S A094384 1,-2,4,16,-32,-128,-512,4096,-8192,-32768,-131072,1048576,4194304,
%T A094384 -33554432,268435456,4294967296,-8589934592,-34359738368,
%U A094384 -137438953472,1099511627776,4398046511104,-35184372088832,281474976710656
%N A094384 Determinant of n X n partial Hadamard matrix with coefficient m(i,j) 1<=i,j<=n (see comment).
%C A094384 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)
%F A094384 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 ?
%F A094384 Conjecture: a(n) = (-2)^A000788(n-1). - _Chai Wah Wu_, Nov 12 2024
%e A094384 M(2)=/1,-1/-1,-1/ then a(2)=detM(2)=-2
%o A094384 (Python)
%o A094384 from sympy import Matrix
%o A094384 def A094384(n):
%o A094384     m = Matrix([1])
%o A094384     for i in range((n-1).bit_length()):
%o A094384         m = Matrix([[m, -m],[-m, -m]])
%o A094384     return m[:n,:n].det() # _Chai Wah Wu_, Nov 12 2024
%Y A094384 Cf. A000788, A073536.
%K A094384 sign
%O A094384 1,2
%A A094384 _Benoit Cloitre_, Jun 03 2004