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

1, 4, 11, 39, 57, 132, 432, 540, 1188, 3780, 4428, 9504, 29808, 33696, 71280, 221616, 244944, 513216, 1586304, 1726272, 3592512, 11057472, 11897280, 24634368, 75582720, 80621568, 166281984, 508923648, 539156736, 1108546560, 3386105856

Offset: 1

Author

Clark Kimberling, Dec 29 2011

Keywords

• nonn
• easy

Comments

From R. J. Mathar, Oct 01 2016 (Start):

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:

1

1 3

1 4 3

1 6 11 6

1 9 29 39 18

1 10 38 68 57 18

1 12 58 144 193 132 36

1 15 94 318 625 711 432 108

1 16 109 412 943 1336 1143 540 108

1 18 141 630 1767 3222 3815 2826 1188 216

1 21 195 1053 3657 8523 13481 14271 9666 3780 648

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

Examples

			Let esf abbreviate "elementary symmetric function".  Then
0th esf of {3}:  1,
1st esf of {3,1}:  3+1=4,
2nd esf of {3,1,2} is 3*1+3*1+1*2=11.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,12,0,0,-36).

Crossrefs

Cf. A010882, A203160, A203162.

Programs

Mathematica

f[k_] := 1 + Mod[k + 1, 3]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 33}] (* 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

Formula

G.f.: x*(3*x+1)*(3*x^3+8*x^2+x+1) / (6*x^3-1)^2. - Colin Barker, Aug 15 2014