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

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Natalia L. Skirrow, bitwise permutations

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)

Example

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

Mathematica

A383718[n_] := Multinomial @@ Map[Length, Split[IntegerDigits[n, 2]]];
Array[A383718, 100, 0] (* Paolo Xausa, Feb 03 2026 *)

Prog

(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)))))

Crossrefs

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

Sequence in context: A006895 A202204 A289815 * A125205 A125206 A221918

Adjacent sequences: A383715 A383716 A383717 * A383719 A383720 A383721

Keyword

nonn,look,base

Author

Natalia L. Skirrow, Apr 20 2025

Status

approved