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%I #16 Dec 30 2023 17:01:08
%S 0,0,1,1,4,3,9,7,15,16,29,23,47,43,74,65,114,100,174,153,257,228,368,
%T 312,530,454,736,645,1025,902,1402,1184,1909,1626,2618,2184,3412,2895,
%U 4551,3887,5966,5055,7796,6509,10244,8462,13060,10881,16834,14021,21471
%N Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.
%H Chai Wah Wu, <a href="/A365323/b365323.txt">Table of n, a(n) for n = 1..95</a>
%e The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
%e The a(3) = 1 through a(10) = 16 partitions:
%e (2) (3) (2) (4) (2) (3) (2) (3)
%e (3) (5) (3) (5) (4) (4)
%e (4) (3,2) (4) (6) (5) (6)
%e (2,2) (5) (7) (6) (7)
%e (6) (3,3) (7) (8)
%e (2,2) (4,3) (8) (9)
%e (3,3) (5,2) (2,2) (3,3)
%e (4,2) (4,2) (4,4)
%e (2,2,2) (4,3) (5,2)
%e (4,4) (5,3)
%e (5,3) (5,4)
%e (6,2) (6,3)
%e (2,2,2) (7,2)
%e (4,2,2) (3,3,3)
%e (2,2,2,2) (4,3,2)
%e (5,2,2)
%t combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]
%o (Python)
%o from sympy.utilities.iterables import partitions
%o def A365323(n):
%o a = {tuple(sorted(set(p))) for p in partitions(n)}
%o return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # _Chai Wah Wu_, Sep 12 2023
%Y Complement for subsets: A088314 or A365042, nonnegative A365073 or A365542.
%Y For strict partitions we have A088528, nonnegative coefficients A365312.
%Y For length-2 subsets we have A365321 (we use n instead of n-1).
%Y For subsets we have A365322 or A365045, nonnegative coefficients A365380.
%Y For nonnegative coefficients we have A365378, complement A365379.
%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 A364350 counts combination-free strict partitions, non-strict A364915.
%Y A364839 counts combination-full strict partitions, non-strict A364913.
%Y Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.
%K nonn
%O 1,5
%A _Gus Wiseman_, Sep 12 2023
%E a(21)-a(51) from _Chai Wah Wu_, Sep 12 2023
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