%I #43 Apr 21 2024 22:11:00
%S 1,2,6,23,106,566,3415,22872,167796,1334596,11414192,104270906,
%T 1011793389,10379989930,112134625986,1271209859403,15077083642150,
%U 186588381229340,2403775013224000,32168379148440968,446341838086450308,6410107231501731012,95136428354649665256
%N a(n) = Sum_{j=0..2^n - 1} b(j) for n >= 0 where b(n) = (A023416(n) + 1)*b(A053645(n)) + [A036987(n) = 0]*b(A266341(n)) for n > 0 with b(0) = 1.
%C Note that [A036987(n) = 0]*b(A266341(n)) is the same as max((1 - T(n, j))*b(A053645(n) + 2^j*(1 - T(n, j))) | 0 <= j <= A000523(n)) where T(n, k) = floor(n/2^k) mod 2.
%C In fact b(n) is a generalization of A347205 just as A329369 is a generalization of A341392.
%o (PARI) A063250(n)=my(L=logint(n, 2), A=0); for(i=0, L, my(B=n\2^(L-i)+1); A++; A-=logint(B, 2)==valuation(B, 2)); A
%o upto(n)=my(v, v1); v=vector(2^n, i, 0); v[1]=1; v1=vector(n+1, i, 0); v1[1]=1; for(i=1, #v-1, my(L=logint(i, 2), A=i - 2^L, B=A063250(i)); v[i+1]=(L - hammingweight(i) + 2)*v[A+1] + if(B>0, v[A + 2^(B-1) + 1])); for(i=1, n, v1[i+1]=v1[i] + sum(j=2^(i-1)+1, 2^i, v[j])); v1
%Y Cf. A000523, A023416, A036987, A053645, A266341.
%Y Similar recurrences: A284005, A329369, A341392, A347205.
%K nonn
%O 0,2
%A _Mikhail Kurkov_, Jun 11 2023 [verification needed]