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.

4, 16, 192, 3072, 61440, 1474560, 41287680, 1321205760, 47563407360, 1902536294400, 83711596953600, 4018156653772800, 208944145996185600, 11700872175786393600, 702052330547183616000, 44931349155019751424000, 3055331742541343096832000

Offset: 1

Author

Clark Kimberling, Mar 05 2025

Keywords

• nonn

Comments

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).

Guide to sequences pos(M(n)), where M(n) is the n X n matrix with every term = c, a constant:

c = 1: A001710

c = 2: A002866

c = 3: A032108

c = 4: this sequence

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.

Crossrefs

Cf. A001710, A002966, A032108, A380661.

Essentially the same as A051711.

Programs

Mathematica

c = 4; d = Table[Det[ConstantArray[c, {n, n}]], {n, 1, 18}]
p = Table[Permanent[ConstantArray[c, {n, n}]], {n, 1, 18}]
neg = (d - p)/2
pos = (d + p)/2

Formula

s(1) = 4, s(2) = 16, and s(n) = 4*n*s(n-1) for n >= 3.