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A357812 Number of subsets of [n] in which exactly half of the elements are powers of 2.

%I A357812 #22 Oct 20 2022 17:35:33
%S A357812 1,1,1,3,4,10,20,35,70,126,210,330,495,715,1001,1365,4368,6188,8568,
%T A357812 11628,15504,20349,26334,33649,42504,53130,65780,80730,98280,118755,
%U A357812 142506,169911,906192,1107568,1344904,1623160,1947792,2324784,2760681,3262623,3838380
%N A357812 Number of subsets of [n] in which exactly half of the elements are powers of 2.
%H A357812 Alois P. Heinz, <a href="/A357812/b357812.txt">Table of n, a(n) for n = 0..10000</a>
%F A357812 a(n) = binomial(n,A029837(n+1)).
%F A357812 a(n) = binomial(n,A113473(n)) for n>0, a(0) = 1.
%F A357812 a(n) = Sum_{j>=0} binomial(A029837(n+1),j)*binomial(n-A029837(n+1),j).
%e A357812 a(6) = 20: {}, {1,3}, {1,5}, {1,6}, {2,3}, {2,5}, {2,6}, {3,4}, {4,5}, {4,6}, {1,2,3,5}, {1,2,3,6}, {1,2,5,6}, {1,3,4,5}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,4,6}, {2,4,5,6}, {1,2,3,4,5,6}.
%p A357812 a:= n-> binomial(n, max(0, 1+ilog[2](n))):
%p A357812 seq(a(n), n=0..40);
%o A357812 (Python)
%o A357812 from math import comb
%o A357812 def A357812(n): return comb(n,n.bit_length()) # _Chai Wah Wu_, Oct 14 2022
%Y A357812 Cf. A000079, A029837, A037031, A102366, A113473, A180272, A357927.
%K A357812 nonn
%O A357812 0,4
%A A357812 _Alois P. Heinz_, Oct 13 2022