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A370819 Number of subsets of {1..n-1} whose cardinality is one less than the length of the binary expansion of n; a(0) = 0.

%I A370819 #6 Mar 13 2024 15:33:11
%S A370819 0,1,1,2,3,6,10,15,35,56,84,120,165,220,286,364,1365,1820,2380,3060,
%T A370819 3876,4845,5985,7315,8855,10626,12650,14950,17550,20475,23751,27405,
%U A370819 169911,201376,237336,278256,324632,376992,435897,501942,575757,658008,749398,850668
%N A370819 Number of subsets of {1..n-1} whose cardinality is one less than the length of the binary expansion of n; a(0) = 0.
%F A370819 a(n) = binomial(n - 1, A029837(n+1) - 1) = binomial(n - 1, A113473(n) - 1) = binomial(n - 1, A070939(n) - 1) for n > 0.
%e A370819 The a(1) = 1 through a(7) = 15 subsets:
%e A370819   {}  {1}  {1}  {1,2}  {1,2}  {1,2}  {1,2}
%e A370819            {2}  {1,3}  {1,3}  {1,3}  {1,3}
%e A370819                 {2,3}  {1,4}  {1,4}  {1,4}
%e A370819                        {2,3}  {1,5}  {1,5}
%e A370819                        {2,4}  {2,3}  {1,6}
%e A370819                        {3,4}  {2,4}  {2,3}
%e A370819                               {2,5}  {2,4}
%e A370819                               {3,4}  {2,5}
%e A370819                               {3,5}  {2,6}
%e A370819                               {4,5}  {3,4}
%e A370819                                      {3,5}
%e A370819                                      {3,6}
%e A370819                                      {4,5}
%e A370819                                      {4,6}
%e A370819                                      {5,6}
%t A370819 Table[If[n==0,0,Binomial[n-1,IntegerLength[n,2]-1]],{n,0,15}]
%Y A370819 The version without subtracting one is A357812.
%Y A370819 Dominates A370641, see also A370640.
%Y A370819 A007318 counts subsets by cardinality.
%Y A370819 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A370819 A058891 counts set-systems, A003465 covering, A323818 connected.
%Y A370819 A070939 gives length of binary expansion.
%Y A370819 A096111 gives product of binary indices.
%Y A370819 Cf. A029837, A113473, A326031, A367905, A368109, A370636.
%K A370819 nonn
%O A370819 0,4
%A A370819 _Gus Wiseman_, Mar 11 2024