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A381724 a(n) = pos(M(n)), where M(n) is the n X n matrix with every term = 4, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.

%I A381724 #9 Mar 11 2025 13:20:54
%S A381724 4,16,192,3072,61440,1474560,41287680,1321205760,47563407360,
%T A381724 1902536294400,83711596953600,4018156653772800,208944145996185600,
%U A381724 11700872175786393600,702052330547183616000,44931349155019751424000,3055331742541343096832000
%N A381724 a(n) = pos(M(n)), where M(n) is the n X n matrix with every term = 4, and pos(M(n)) is the positive part of the determinant of M(n); see A380661.
%C A381724 For a matrix M with determinant |M|, the numbers pos(M) and neg(M) are the positive and negative parts of |M|, as defined in A380661. The definition implies that (pos(M)+neg(M))/2 = |M| and (pos(M)-neg(M))/2 = permanent of M. Thus, M is singular if and only if pos(M) = - neg(M).
%C A381724 Guide to sequences pos(M(n)), where M(n) is the n X n matrix with every term = c, a constant:
%C A381724   c = 1:  A001710
%C A381724   c = 2:  A002866
%C A381724   c = 3:  A032108
%C A381724   c = 4:  this sequence
%C A381724 For each c >=1, let s(n) = pos(M(n)); then s(1) = c, s(2) = c^2, and s(n) = c*n*s(n-1) for n >= 3.
%F A381724 s(1) = 4, s(2) = 16, and s(n) = 4*n*s(n-1) for n >= 3.
%t A381724 c = 4; d = Table[Det[ConstantArray[c, {n, n}]], {n, 1, 18}]
%t A381724 p = Table[Permanent[ConstantArray[c, {n, n}]], {n, 1, 18}]
%t A381724 neg = (d - p)/2
%t A381724 pos = (d + p)/2
%Y A381724 Cf. A001710, A002966, A032108, A380661.
%Y A381724 Essentially the same as A051711.
%K A381724 nonn
%O A381724 1,1
%A A381724 _Clark Kimberling_, Mar 05 2025