A305188 - OEIS

%I #105 Jul 23 2020 03:27:17

%S 6,10,12,15,20,21,24,28,30,35,36,42,45,55,56,60,66,70,72,78,84,90,91,

%T 105,110,120,126,132,136,140,153,156,165,168,171,180,182,190,210,220,

%U 231,240,252,253,272,276,280,286,300,306,325,330,336,342,351,360,364

%N Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!*...*km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km).

%C This sequence answers the following question: what numbers correspond to the number of permutations of a number of items that is lower than the number of permutations itself? It means that the underlying structure has some form of redundance / symmetry.

%C It can be shown that:

%C - no prime number is part of this sequence (see A304938)

%C - some nonprimes are not part of the sequence (beginning with 1, 4, 8, 9, 14, 16, 18, ...)

%C - any number that is a factorial of another integer e is part of this sequence (k=e, k1=ki=...=ke=1).

%C This sequence is a generalization of A006987.

%C From _David A. Corneth_, May 28 2018: (Start)

%C Also the numbers that are the number of permutations of either:

%C - sets of balls with two distinct colors of balls where each color occurs at least twice;

%C - sets of balls with at least three distinct colors of balls.

%C (End)

%C From _Amiram Eldar_, Jul 23 2020: (Start)

%C The asymptotic density of this sequence is 0 (Niven, 1951).

%C The number of terms not exceeding x is (1 + sqrt(2)) * x^(1/2) + o(x^(1/2)) (Erdős, 1954). (End)

%H David A. Corneth, <a href="/A305188/b305188.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincent Champain, <a href="https://repl.it/@vcch/OEIS305188">Calculation of the terms of the sequence including a k1..km list so that a(n) = k!/(k1!*...*km!) with k1+...+km = k and m > 1</a>, 2018.

%H Vincent Champain, <a href="/A305188/a305188.txt">Python program for A305188</a>

%H David A. Corneth, <a href="/A305188/a305188.gp.txt">Terms with the corresponding tuples. </a>

%H Paul Erdős, <a href="http://www.jstor.org/stable/2306893">The Number of Multinomial Coefficients</a>, The American Mathematical Monthly, Vol. 61, No. 1 (1954), pp. 37-39.

%H Ivan Niven, <a href="https://doi.org/10.1090/S0002-9904-1951-09543-9">The asymptotic density of sequences</a>, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. See theorem 2, p. 428.

%e a(1) = 6 because all numbers lower than 6 are either prime or a power of primes.

%e 105 is a term of the sequence because 105 is equal to a multinomial coefficient: 105 = (4+2+1)! / (4! * 2! * 1!) and 105 is the number of ways 7 balls can be sorted where 4 are red, 2 are yellow and one is blue.

%e 2016 is a term because 64! / (62! * 2!) = 2016. - _David A. Corneth_, May 29 2018

%t mult[w_] := Total[w]!/Times @@ (w!); L = {}; Do[ t = mult /@ Select[ IntegerPartitions@ n, #[[1]] < n-1 &]; L = Union[L, Select[t, # <= 400 &]], {n, 3, 30}]; L (* Terms < 400, _Giovanni Resta_, May 27 2018 *)

%o (Python) # see link above

%Y Cf. A008480, A304938, A006987.

%K nonn

%O 1,1

%A _Vincent Champain_, May 27 2018

%E a(28)-a(57) from _Giovanni Resta_, May 27 2018