cp's OEIS Frontend

A365542 Number of subsets of {1..n-1} that can be linearly combined using nonnegative coefficients to obtain n.

%I A365542 #13 Sep 13 2023 08:35:52
%S A365542 0,1,2,6,10,28,48,116,224,480,920,2000,3840,7984,15936,32320,63968,
%T A365542 130176,258304,521920,1041664,2089472,4171392,8377856,16726528,
%U A365542 33509632,67004416,134129664,268111360,536705024,1072961536,2146941952,4293509120,8588414976
%N A365542 Number of subsets of {1..n-1} that can be linearly combined using nonnegative coefficients to obtain n.
%e A365542 The a(2) = 1 through a(5) = 10 partitions:
%e A365542   {1}  {1}    {1}      {1}
%e A365542        {1,2}  {2}      {1,2}
%e A365542               {1,2}    {1,3}
%e A365542               {1,3}    {1,4}
%e A365542               {2,3}    {2,3}
%e A365542               {1,2,3}  {1,2,3}
%e A365542                        {1,2,4}
%e A365542                        {1,3,4}
%e A365542                        {2,3,4}
%e A365542                        {1,2,3,4}
%t A365542 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365542 Table[Length[Select[Subsets[Range[n-1]],combs[n,#]!={}&]],{n,5}]
%o A365542 (Python)
%o A365542 from itertools import combinations
%o A365542 from sympy.utilities.iterables import partitions
%o A365542 def A365542(n):
%o A365542     a = {tuple(sorted(set(p))) for p in partitions(n)}
%o A365542     return sum(1 for m in range(1,n) for b in combinations(range(1,n),m) if any(set(d).issubset(set(b)) for d in a)) # _Chai Wah Wu_, Sep 12 2023
%Y A365542 The case of positive coefficients is A365042, complement A365045.
%Y A365542 For subsets of {1..n} instead of {1..n-1} we have A365073.
%Y A365542 The binary complement is A365315.
%Y A365542 The complement is counted by A365380.
%Y A365542 A124506 and A326083 appear to count combination-free subsets.
%Y A365542 A179822 and A326080 count sum-closed subsets.
%Y A365542 A364350 counts combination-free strict partitions.
%Y A365542 A364914 and A365046 count combination-full subsets.
%Y A365542 Cf. A007865, A088314, A088809, A151897, A364534, A364839, A365314, A365322.
%K A365542 nonn
%O A365542 1,3
%A A365542 _Gus Wiseman_, Sep 09 2023
%E A365542 More terms from _Alois P. Heinz_, Sep 13 2023