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%I #17 May 01 2022 14:02:59
%S 4,6,13,43,225,1505,11177,86745,684889
%N Number of distinct terms by nesting the 2nd elementary symmetric polynomial in 4 variables.
%e 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.
%p s:= proc(n) option remember; `if`(n=1, x+y+z+w,
%p (p-> expand(p^2-map(u->u^2, p)))(s(n-1)))
%p end:
%p a:= n-> nops(s(n)):
%p seq(a(n), n=1..7); # _Alois P. Heinz_, Sep 02 2014
%t With[{g = Level[SymmetricPolynomial[2, #], 1] &},
%t Length /@ NestList[g, {x, y, z, w}, 5]]
%t (* Second program: *)
%t s[n_] := s[n] = If[n == 1, x + y + z + w,
%t With[{p = s[n-1]}, Expand[p^2 - (#^2& /@ p)]]];
%t a[n_] := Length[s[n]];
%t Table[a[n], {n, 1, 7}] (* _Jean-François Alcover_, May 01 2022, after _Alois P. Heinz_ *)
%K nonn,hard,more
%O 1,1
%A _Filippo Miatto_, Aug 26 2014
%E a(8) from _Alois P. Heinz_, Sep 02 2014
%E a(9) from _Alois P. Heinz_, Oct 06 2014
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