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 A383718 #11 Jun 02 2025 16:49:53
%S A383718 1,1,2,1,3,6,3,1,4,12,24,12,6,12,4,1,5,20,60,30,60,120,60,20,10,30,60,
%T A383718 30,10,20,5,1,6,30,120,60,180,360,180,60,120,360,720,360,180,360,120,
%U A383718 30,15,60,180,90,180,360,180,60,20,60,120,60,15,30,6,1
%N A383718 a(n) is the multinomial coefficient (length of n in binary) choose (the lengths of runs in n's binary expansion).
%H A383718 Natalia L. Skirrow, <a href="https://oeis.org/wiki/User:Nathan_L._Skirrow/bitwise_permutations">bitwise permutations</a>
%F A383718 a(n) >= A368070(n), with equality iff n is in A023758. (In particular, if n' is formed by appending a bit to n's expansion, a(n')/A368070(n') >= a(n)/A368070(n).)
%F A383718 The ratio r = a(n)/A368070(n) reaches minima when n is in A000975; a(A000975(n)) = n!, whereas A368070(A000975(n)) = A000111(n+1).
%F A383718 As such, lim inf r = 0, but lim inf_{n>=m} log(a(n))/log(A368070(n)) is 1, converging as about 1 - log_{log_2(n)}(Pi/2)
%e A383718 2025_10 = 11111101001_2, with run lengths {6,1,1,2,1}; 11!/(6!*1!^3*2!) = 27720.
%o A383718 (Python)
%o A383718 from itertools import groupby
%o A383718 from math import prod, factorial as fact
%o A383718 rlenomial=lambda n: fact(l:=n.bit_length())//prod(map(lambda n: fact(len(list(n[1]))),groupby(map(lambda i: n>>i&1,range(l)))))
%Y A383718 Cf. A101211, A368070, A023758, A000975, A000111.
%K A383718 nonn,base
%O A383718 0,3
%A A383718 _Natalia L. Skirrow_, Apr 20 2025