1,2

A sequence generated from an inverse Bell matrix.

a(n)/a(n-1) tends to 3.4908636153...; a root of x^3 - 3*x^2 - 2x + 1 and an eigenvalue of M. A095127 is generated from the reflected polynomial: x^3 - 2x^2 - 3x + 1 and the inverse matrix of M. Bell numbers are sums of row terms of the 3rd order Stirling number of the second kind matrix shown on p. 171 of Aldrovandi, the matrix being [ 1 0 0 / 1 1 0 / 1 3 1]. Rotations, or inverses, or related polynomials generate A095125, A095126, A095127, A095128.

R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, section 13.3.1, "Inverting Bell Matrices", p. 171.

Invert the matrix used to generate A095127, getting M = [3 2 -1 / 1 0 0 / 0 1 0]. Then M^n * [1 1 1] = [p q r] where a(n) = the center term q.

a(6) = 559 = 3*a(5) + 2*a(4) - a(3) = 3*160 + 2*46 - 13.

a(4) = 46 since M^4 * [1 1 1] = [160 46 13] = [a(5) a(4) a(3)].

a[n_] := (MatrixPower[{{3, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 24}] (from Robert G. Wilson v Jun 01 2004)

Cf. A095125, A095126, A095127.

Sequence in context: A096353 A034553 A104460 this_sequence A047154 A026641 A087440

Adjacent sequences: A095125 A095126 A095127 this_sequence A095129 A095130 A095131

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), May 29 2004

Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 01 2004