A383718 a(n) is the multinomial coefficient (length of n in binary) choose (the lengths of runs in n's binary expansion).

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, 30, 10, 20, 5, 1, 6, 30, 120, 60, 180, 360, 180, 60, 120, 360, 720, 360, 180, 360, 120, 30, 15, 60, 180, 90, 180, 360, 180, 60, 20, 60, 120, 60, 15, 30, 6, 1

Offset: 0

Author

Natalia L. Skirrow, Apr 20 2025

Keywords

• nonn
• base

Examples

			2025_10 = 11111101001_2, with run lengths {6,1,1,2,1}; 11!/(6!*1!^3*2!) = 27720.
		

Links

• Natalia L. Skirrow, bitwise permutations

Crossrefs

Cf. A101211, A368070, A023758, A000975, A000111.

Programs

Python

from itertools import groupby
from math import prod, factorial as fact
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)))))

Formula

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).)

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).

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)