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A095126 Expansion of x*(4+5*x-x^2)/ (1-2*x-3*x^2+x^3).

%I A095126 #21 Jan 05 2025 23:38:34
%S A095126 4,13,37,109,316,922,2683,7816,22759,66283,193027,562144,1637086,
%T A095126 4767577,13884268,40434181,117753589,342925453,998677492,2908377754,
%U A095126 8469862531,24666180832,71833571503,209195822971,609226179619
%N A095126 Expansion of x*(4+5*x-x^2)/ (1-2*x-3*x^2+x^3).
%C A095126 A sequence generated from a rotated Stirling number of the second kind matrix, companion to A095125.
%C A095126 a(n)/a(n-1) tends to 2.9122291784...an eigenvalue of M and a root of the characteristic polynomial x^3 - 2x^2 - 3x + 1. A095127 is generated from the same polynomial, with the reversal x^3 - 3x^2 - 2x + 1 being the characteristic polynomial of A095128.
%D A095126 R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, Section 13.3.1, "Inverting Bell Matrices", p. 171.
%H A095126 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-1).
%F A095126 a(n+3) = 2*a(n+2) + 3*a(n+1) - a(n); with a(1) = 4, a(2) = 13, a(3) = 37.
%F A095126 Let M = a rotated Stirling number of the second kind matrix [1 1 1 / 3 1 0 / 1 0 0] (a rotation of [1 0 0 / 1 1 0 / 1 3 1]). Then M^n * [1 1 1] = [A095125(n+1), a(n), A095125(n)].
%e A095126 a(6) = 922 = 2*316 + 3*109 - 37 = 2*a(5) + 3*a(4) - a(3).
%e A095126 a(5) = 316 since M^5 * [1 1 1] = [202 316 69] = [A095125(6), a(n), A095125(5)]
%t A095126 a[n_] := (MatrixPower[{{1, 1, 1}, {3, 1, 0}, {1, 0, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 26}] (* _Robert G. Wilson v_, Jun 01 2004 *)
%t A095126 LinearRecurrence[{2,3,-1},{4,13,37},30] (* _Harvey P. Dale_, Jan 18 2016 *)
%o A095126 (PARI) Vec((4+5*x-x^2)/(1-2*x-3*x^2+x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y A095126 Cf. A095125, A095127, A095128.
%K A095126 nonn,easy
%O A095126 1,1
%A A095126 _Gary W. Adamson_, May 29 2004
%E A095126 Edited, corrected and extended by _Robert G. Wilson v_, Jun 01 2004