A203162 (n-1)-st elementary symmetric function of the first n terms of (1,2,3,1,2,3,1,2,3,...).

1, 3, 11, 17, 40, 132, 168, 372, 1188, 1404, 3024, 9504, 10800, 22896, 71280, 79056, 165888, 513216, 559872, 1166400, 3592512, 3872448, 8024832, 24634368, 26313984, 54307584, 166281984, 176359680, 362797056, 1108546560, 1169012736

Offset: 1

Author

Clark Kimberling, Dec 29 2011

Keywords

• nonn
• easy

Examples

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

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,12,0,0,-36).

Crossrefs

Cf. A010882, A203160, A203161.

Programs

GAP

a:=[1, 3, 11, 17, 40, 132];; for n in [7..40] do a[n]:=12*a[n-1]-36*a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 10 2019

Magma

I:=[1, 3, 11, 17, 40, 132]; [n le 6 select I[n] else 12*Self(n-3) -36*Self(n-6): n in [1..40]]; // G. C. Greubel, May 10 2019

Mathematica

f[k_] := 1 + Mod[k + 2, 3]; t[n_] := Table[f[k], {k, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 40}] (* A203162 *)
Rest[CoefficientList[Series[x*(1 + 3*x + 11*x^2 + 5*x^3 + 4*x^4)/(1 - 6*x^3)^2, {x, 0, 30}], x]] (* Vaclav Kotesovec, May 10 2019 *)

PARI

my(x='x+O('x^40)); Vec(x*(1+3*x+11*x^2+5*x^3+4*x^4)/(1-6*x^3)^2) \\ G. C. Greubel, May 10 2019

Sage

a=(x*(1+3*x+11*x^2+5*x^3+4*x^4)/(1-6*x^3)^2).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 10 2019

Formula

a(n) = 12*a(n-3)-36*a(n-6). - Clark Kimberling, Aug 18 2012

G.f.: x*(1 + 3*x + 11*x^2 + 5*x^3 + 4*x^4)/(1 - 6*x^3)^2. - Clark Kimberling, Aug 18 2012; corrected by Georg Fischer, May 10 2019