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A196534 Number of different ways to select disjoint nonempty subsets from {1..n} with equal element sum.

%I A196534 #16 Jun 01 2022 08:52:56
%S A196534 1,3,8,18,39,83,179,388,857,1914,4494,10844,26923,70645,192297,538646,
%T A196534 1579602,4793718,15010425,48941642,164010913,566065123,2025354291,
%U A196534 7450901462,27986863322,107940691328
%N A196534 Number of different ways to select disjoint nonempty subsets from {1..n} with equal element sum.
%C A196534 A000225(n) <= a(n) <= A058692(n+1).
%e A196534 a(3) = 8: {{1}}, {{2}}, {{3}}, {{1,2}}, {{1,3}}, {{2,3}}, {{1,2,3}}, {{1,2},{3}}. Element sums are 1, 2, 3, 3, 4, 5, 6, and 3, respectively.
%p A196534 b:= proc(l, n, k) option remember; local i, j; `if`(l=[0$k], 1, `if`(add(j, j=l)>n*(n-1)/2, 0, b(l, n-1, k))+ add(`if`(l[j]-n<0, 0, b(sort([seq(l[i] -`if`(i=j, n, 0), i=1..k)]), n-1, k)), j=1..k)) end: a:= n-> add(add(b([t$k], n, k), t=2*k-1..floor(n*(n+1)/(2*k)))/k!, k=1..n): seq(a(n), n=1..15);
%t A196534 b[l_, n_, k_] := b[l, n, k] = If[l == Array[0&, k], 1, If[Total[l] > n*(n-1)/2, 0, b[l, n-1, k]] + Sum[If[l[[j]]-n < 0, 0, b[Sort[Table[ l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}]];
%t A196534 a[n_] := Sum[Sum[b[Array[t&, k], n, k], {t, 2*k-1, Floor[n*(n+1)/(2*k)]} ]/k!, {k, 1, Ceiling[n/2]}];
%t A196534 Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* _Jean-François Alcover_, Jun 01 2022, after _Alois P. Heinz_ *)
%Y A196534 Row sums of A196231.  Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236, A196237, A058692.
%K A196534 nonn,more
%O A196534 1,2
%A A196534 _Alois P. Heinz_, Oct 03 2011
%E A196534 a(26) from _Alois P. Heinz_, Oct 20 2014