%I #16 Dec 30 2023 21:23:13
%S 0,0,0,1,2,4,7,10,16,23,34,44,67,85,119,157,210,268,360,453,592,748,
%T 956,1195,1520,1883,2365,2920,3628,4451,5494,6702,8211,9976,12147,
%U 14666,17776,21389,25774,30887,37035,44224,52819,62836,74753,88614,105062,124160
%N Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.
%C These may be called "non-binary nonnegative combination-full" partitions.
%C Does not necessarily include all non-strict partitions (A047967).
%e The partition (5,4,3,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(15).
%e The partition (6,4,3,2) has 6 = 1*2 + 1*4, so is counted under a(15). The combinations 6 = 2*3 = 3*2 and 4 = 2*2 can also be used.
%e The a(3) = 1 through a(8) = 16 partitions:
%e (21) (31) (41) (42) (61) (62)
%e (211) (221) (51) (331) (71)
%e (311) (321) (421) (422)
%e (2111) (411) (511) (431)
%e (2211) (2221) (521)
%e (3111) (3211) (611)
%e (21111) (4111) (3221)
%e (22111) (3311)
%e (31111) (4211)
%e (211111) (5111)
%e (22211)
%e (32111)
%e (41111)
%e (221111)
%e (311111)
%e (2111111)
%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
%t Table[Length[Select[IntegerPartitions[n], Function[ptn,Or@@Table[combs[ptn[[k]], DeleteCases[ptn,ptn[[k]]]]!={}, {k,Length[ptn]}]]]],{n,0,5}]
%o (Python)
%o from sympy.utilities.iterables import partitions
%o def A365068(n):
%o if n <= 1: return 0
%o alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
%o for p in partitions(n,k=n-1):
%o s = set(p)
%o if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
%o c += 1
%o return c # _Chai Wah Wu_, Sep 20 2023
%Y The complement for sums instead of combinations is A237667, binary A236912.
%Y For sums instead of combinations we have A237668, binary A237113.
%Y The strict case is A364839, complement A364350.
%Y Allowing equal parts in the combination gives A364913.
%Y For subsets instead of partitions we have A364914, complement A326083.
%Y The complement is A364915.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A116861 and A364916 count linear combinations of strict partitions.
%Y A323092 counts double-free partitions, ranks A320340.
%Y A364912 counts linear combinations of partitions of k.
%Y Cf. A108917, A151897, A364272, A364910, A364911, A365006.
%K nonn
%O 0,5
%A _Gus Wiseman_, Aug 27 2023
%E a(31)-a(47) from _Chai Wah Wu_, Sep 20 2023