This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380944 #4 Feb 12 2025 12:57:22
%S A380944 1,1,2,1,3,2,3,1,4,3,5,2,9,3,4,1,5,4,7,3,13,5,7,2,23,9,12,3,16,4,5,1,
%T A380944 6,5,9,4,17,7,10,3,31,13,18,5,25,7,9,2,53,23,32,9,44,12,15,3,64,16,20,
%U A380944 4,25,5,6,1,7,6,11,5,21,9,13,4,39,17,24,7,34,10
%N A380944 a(n) = b(n,A000120(n)) for n >= 0 where b(n,k) is defined in Comments.
%C A380944 Here b(2n+1,k) = b(n,k) + b(n,k-1) for n >= 0, k > 0, b(2n,k) = (A000120(n)-k+1)*b(2n+1,k) for n > 0, k > 0 with T(n, 0) = 1 for n >= 0, T(0, k) = 0 for k > 0 (see A379817, A379819 for similar recurrence).
%F A380944 Conjecture: a(2^m*(2^n-1)) = (n+1)^m - n*n!*c(m,n+1) for n >= 0, m >= 0 where c(n,k) = Sum_{i=0..n-k} Stirling2(k+i,k) for n >= 0, k >= 0.
%o A380944 (PARI) b(n,k) = if(k==0, 1, if(n==0, 0, if(n%2, b((n-1)/2,k) + b((n-1)/2,k-1), (hammingweight(n/2)-k+1)*b(n+1,k))))
%o A380944 a(n) = b(n, hammingweight(n))
%Y A380944 Cf. A000120, A379817, A379819.
%K A380944 nonn,base
%O A380944 0,3
%A A380944 _Mikhail Kurkov_, Feb 09 2025