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%I #16 Jan 14 2024 00:15:00
%S 0,1,1,1,1,2,1,2,1,2,2,3,1,3,2,2,2,3,2,4,2,2,3,4,1,4,3,3,3,4,3,4,2,3,
%T 5,5,1,4,4,4,5,4,2,6,3,3,6,5,3,5,4,4,4,6,4,8,2,2,8,6,3,5,6,4,6,6,3,7,
%U 4,5,9,6,3,6,6,6,7,4,5,9,5,3,9,9,3,7,6,4,10,8
%N Number of sets S of positive integers satisfying E(S)=n, where E = 2nd elementary symmetric function.
%H Andrew Howroyd, <a href="/A026904/b026904.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial">Elementary symmetric polynomial</a>
%e a(2) = 2 counts {1,11}, {1,2,3}.
%t a[n_] := Module[{r}, r[lim_, s1_, s2_] := r[lim, s1, s2] = If[s2 == n, 1, Sum[r[i, s1 + i, s2 + s1*i], {i, 1, Min[Quotient[n-s2, s1], lim - 1]}]]; Sum[r[i, i, 0], {i, 1, n}]];
%t a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 28 2019, after _Andrew Howroyd_ *)
%o (PARI) a(n)={my(recurse(lim, s1, s2)=if(s2==n, 1, sum(i=1, min((n-s2)\s1, lim-1), self()(i, s1+i, s2+s1*i)))); sum(i=1, n, recurse(i, i, 0))} \\ _Andrew Howroyd_, Dec 17 2018
%Y Cf. A026903.
%K nonn
%O 1,6
%A _Clark Kimberling_
%E Terms a(35) and beyond from _Andrew Howroyd_, Dec 17 2018
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