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%I #25 Sep 28 2019 08:37:03
%S 1,1,2,2,2,3,2,3,4,3,3,5,3,4,6,4,4,6,4,6,7,4,5,9,6,6,8,6,7,9,5,9,10,6,
%T 9,12,6,8,12,9,9,12,8,11,14,8,10,15,11,12,13,10,13,16,11,15,16,8,14,
%U 21,12,14,19,14,16,18,12,17,21,14,17,23,15,19,22
%N a(n) is the number of multisets S of positive integers satisfying E(S)=n, where E = 2nd elementary symmetric function.
%H Andrew Howroyd, <a href="/A026903/b026903.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(9) = 4 counts {1,9}, {3,3}, {1,1,4}, {1,1,1,2}.
%e a(33) = 10 counts {1,1,1,1,1,1,3}, {1,1,1,2,2,2}, {1,1,1,10}, {1,1,2,7}, {1,1,4,4}, {1,1,16}, {1,2,2,5}, {1,33}, {3,3,4}, {3,11}.
%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]}]]; 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), self()(i, s1+i, s2+s1*i)))); sum(i=1, n, recurse(i, i, 0))} \\ _Andrew Howroyd_, Dec 17 2018
%Y Cf. A026904.
%K nonn
%O 1,3
%A _Clark Kimberling_
%E a(33) corrected by _Laurance L. Y. Lau_, Dec 17 2018
%E Terms a(35) and beyond from _Andrew Howroyd_, Dec 17 2018
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