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A203161 (n-1)-st elementary symmetric function of the first n terms of (3,1,2,3,1,2,3,1,2,...).

%I A203161 #17 Nov 28 2017 11:35:53
%S A203161 1,4,11,39,57,132,432,540,1188,3780,4428,9504,29808,33696,71280,
%T A203161 221616,244944,513216,1586304,1726272,3592512,11057472,11897280,
%U A203161 24634368,75582720,80621568,166281984,508923648,539156736,1108546560,3386105856
%N A203161 (n-1)-st elementary symmetric function of the first n terms of  (3,1,2,3,1,2,3,1,2,...).
%C A203161 From _R. J. Mathar_, Oct 01 2016 (Start):
%C A203161 The k-th elementary symmetric functions of the first n terms of 3,1,2,3,1,2.., form a triangle T(n,k), 0<=k<=n, n>=0:
%C A203161 1
%C A203161 1 3
%C A203161 1 4 3
%C A203161 1 6 11 6
%C A203161 1 9 29 39 18
%C A203161 1 10 38 68 57 18
%C A203161 1 12 58 144 193 132 36
%C A203161 1 15 94 318 625 711 432 108
%C A203161 1 16 109 412 943 1336 1143 540 108
%C A203161 1 18 141 630 1767 3222 3815 2826 1188 216
%C A203161 1 21 195 1053 3657 8523 13481 14271 9666 3780 648
%C A203161 This here is the first subdiagonal. The diagonal is a stuttered version of A026532. The 2nd column is A047231 (or A144429). (End)
%H A203161 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,12,0,0,-36).
%F A203161 G.f.: x*(3*x+1)*(3*x^3+8*x^2+x+1) / (6*x^3-1)^2. - _Colin Barker_, Aug 15 2014
%e A203161 Let esf abbreviate "elementary symmetric function".  Then
%e A203161 0th esf of {3}:  1,
%e A203161 1st esf of {3,1}:  3+1=4,
%e A203161 2nd esf of {3,1,2} is 3*1+3*1+1*2=11.
%t A203161 f[k_] := 1 + Mod[k + 1, 3]; t[n_] := Table[f[k], {k, 1, n}]
%t A203161 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A203161 Table[a[n], {n, 1, 33}] (* A203161 *)
%o A203161 (PARI) Vec(x*(3*x+1)*(3*x^3+8*x^2+x+1)/(6*x^3-1)^2 + O(x^100)) \\ _Colin Barker_, Aug 15 2014
%Y A203161 Cf. A010882, A203160, A203162.
%K A203161 nonn,easy
%O A203161 1,2
%A A203161 _Clark Kimberling_, Dec 29 2011