A246424 Number of distinct terms by nesting the 2nd elementary symmetric polynomial in 4 variables.

4, 6, 13, 43, 225, 1505, 11177, 86745, 684889

Offset

1,1

Links

Table of n, a(n) for n=1..9.

Example

The second elementary symmetric polynomial s_2 of 4 variables (x,y,z,w) is s_2(x,y,z,w) = xy + xz + xw + yz + yw + zw, so it has 6 distinct terms. Now we take each of those terms and consider it a new variable and calculate s_2 again. This time it will have 13 distinct terms (15 in total, of which 3 are the same). Repeat, you will find 43 distinct terms, and so on.

Maple

s:= proc(n) option remember; `if`(n=1, x+y+z+w,
      (p-> expand(p^2-map(u->u^2, p)))(s(n-1)))
    end:
a:= n-> nops(s(n)):
seq(a(n), n=1..7);  # Alois P. Heinz, Sep 02 2014

Mathematica

With[{g = Level[SymmetricPolynomial[2, #], 1] &},
Length /@ NestList[g, {x, y, z, w}, 5]]
(* Second program: *)
s[n_] := s[n] = If[n == 1, x + y + z + w,
     With[{p = s[n-1]}, Expand[p^2 - (#^2& /@ p)]]];
a[n_] := Length[s[n]];
Table[a[n], {n, 1, 7}] (* Jean-François Alcover, May 01 2022, after Alois P. Heinz *)

Crossrefs

Sequence in context: A302311 A160805 A359020 * A012776 A240198 A016072

Adjacent sequences: A246421 A246422 A246423 * A246425 A246426 A246427

Keyword

nonn,hard,more

Author

Filippo Miatto, Aug 26 2014

Extensions

a(8) from Alois P. Heinz, Sep 02 2014

a(9) from Alois P. Heinz, Oct 06 2014

Status

approved