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A061095 Number of ways of dividing n labeled items into labeled boxes with an equal number of items in each box. 14
1, 3, 7, 31, 121, 831, 5041, 42911, 364561, 3742453, 39916801, 486891175, 6227020801, 87859375033, 1307843292757, 21004582611871, 355687428096001, 6415015584161757, 121645100408832001, 2435278206317164781, 51091124681475552961, 1124549556257968545433 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..449

FORMULA

a(n) = Sum_{d|n} n!/(n/d)!^d.

E.g.f.: Sum_{k>0} x^k/(k!-x^k). - Vladeta Jovovic, Oct 13 2003

a(n) ~ n!. - Vaclav Kotesovec, Jul 21 2019

EXAMPLE

a(6) = 720+90+20+1 = 831 since 720 ways of evenly distributing six labeled items into six labeled boxes, 90 into three, 20 into two and 1 into one.

MAPLE

A061095 := n -> add(n!/(n/d)!^d, d = numtheory[divisors](n));

seq(A061095 (n), n = 1..17); # Peter Luschny, Apr 13 2011

MATHEMATICA

Table[Sum[n!/(n/d)!^d, {d, Divisors[n]}], {n, 1, 20}]  (* Geoffrey Critzer, Aug 18 2011 *)

PROG

(PARI) /* compare to A038041 */

mnom(v)=

/* Multinomial coefficient s! / prod(j=1, n, v[j]!) where

  s= sum(j=1, n, v[j]) and n is the number of elements in v[]. */

sum(j=1, #v, v[j])! / prod(j=1, #v, v[j]!)

A061095(n)={local(r=0); fordiv(n, d, r+=mnom(vector(d, j, n/d))); return(r); }

vector(33, n, A061095(n)) /* Joerg Arndt, Apr 16 2011 */

(PARI) a(n)=sumdiv(n, d, n!/(n/d)!^d ); \\ Joerg Arndt, Feb 23 2014

CROSSREFS

Cf. A038041, A060538.

Column k=1 of A327803.

Sequence in context: A173062 A306831 A105765 * A103901 A089162 A016047

Adjacent sequences:  A061092 A061093 A061094 * A061096 A061097 A061098

KEYWORD

nonn,nice,easy

AUTHOR

Henry Bottomley, May 29 2001

STATUS

approved

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Last modified February 28 08:20 EST 2020. Contains 332321 sequences. (Running on oeis4.)