A203161 - OEIS

%I #17 Nov 28 2017 11:35:53

%S 1,4,11,39,57,132,432,540,1188,3780,4428,9504,29808,33696,71280,

%T 221616,244944,513216,1586304,1726272,3592512,11057472,11897280,

%U 24634368,75582720,80621568,166281984,508923648,539156736,1108546560,3386105856

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

%C From _R. J. Mathar_, Oct 01 2016 (Start):

%C 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 1

%C 1 3

%C 1 4 3

%C 1 6 11 6

%C 1 9 29 39 18

%C 1 10 38 68 57 18

%C 1 12 58 144 193 132 36

%C 1 15 94 318 625 711 432 108

%C 1 16 109 412 943 1336 1143 540 108

%C 1 18 141 630 1767 3222 3815 2826 1188 216

%C 1 21 195 1053 3657 8523 13481 14271 9666 3780 648

%C This here is the first subdiagonal. The diagonal is a stuttered version of A026532. The 2nd column is A047231 (or A144429). (End)

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,12,0,0,-36).

%F 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 Let esf abbreviate "elementary symmetric function".  Then

%e 0th esf of {3}:  1,

%e 1st esf of {3,1}:  3+1=4,

%e 2nd esf of {3,1,2} is 3*1+3*1+1*2=11.

%t f[k_] := 1 + Mod[k + 1, 3]; t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[n - 1, t[n]]

%t Table[a[n], {n, 1, 33}] (* A203161 *)

%o (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 Cf. A010882, A203160, A203162.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Dec 29 2011