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%I #21 Jan 20 2023 01:31:37
%S 0,0,0,0,2,9,28,74,178,402,872,1842,3821,7830,15913,32161,64761,
%T 130091,260911,522749,1046667,2094797,4191414,8385079,16772926,
%U 33549239,67102603,134210207,268426453,536860171,1073729049,2147468499,4294949383,8589913467,17179844335
%N Number of nonempty subsets S of {1,2,...,n} in which all elements are strictly less than the sum of the other elements of S.
%C In other words, every element of S is strictly less than half the sum.
%H math.stackexchange, <a href="https://math.stackexchange.com/a/4024703/6460">Permutation and Combinatorics Problem</a>
%F a(n) = A095941(n) - A317910(n).
%F G.f.: (1-x-x^2)/((1-x)^2*(1-2*x)) - (1/(1-x)^2)*Product_{k>=1} (1 + x^k).
%e For n = 5 the a(5)=9 subsets are {2,3,4}, {2,4,5}, {3,4,5}, {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, and {1,2,3,4,5}.
%p b:= proc(n, s) option remember; `if`(s<1, 2^n,
%p `if`(n*(n+1)/2<s, 0, b(n-1, s)+b(n-1, max(0, s-n))))
%p end:
%p a:= n-> add(b(j-1, j+1), j=1..n):
%p seq(a(n), n=0..37); # _Alois P. Heinz_, Feb 13 2021
%t gf := (1 - x - x^2)/((1 - 2 x) (1 - x)^2) - QPochhammer[-1, x]/(2 (1 - x)^2);
%t CoefficientList[Series[gf, {x, 0, 34}], x] (* _Peter Luschny_, Feb 13 2021 *)
%Y Cf. A000009, A002623, A095941, A095944, A317910.
%K nonn
%O 0,5
%A _Henry Bottomley_, Feb 13 2021
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