cp's OEIS Frontend

A061095 Number of ways of dividing n labeled items into labeled boxes with an equal number of items in each box.

%I A061095 #35 May 02 2023 16:24:58
%S A061095 1,3,7,31,121,831,5041,42911,364561,3742453,39916801,486891175,
%T A061095 6227020801,87859375033,1307843292757,21004582611871,355687428096001,
%U A061095 6415015584161757,121645100408832001,2435278206317164781,51091124681475552961,1124549556257968545433
%N A061095 Number of ways of dividing n labeled items into labeled boxes with an equal number of items in each box.
%H A061095 Seiichi Manyama, <a href="/A061095/b061095.txt">Table of n, a(n) for n = 1..449</a>
%H A061095 Kévin Perrot, Sylvain Sené, and Léah Tapin, <a href="https://arxiv.org/abs/2304.09664">On countings and enumerations of block-parallel automata networks</a>, arXiv:2304.09664 [cs.DM], 2023.
%F A061095 a(n) = Sum_{d|n} n!/(n/d)!^d.
%F A061095 E.g.f.: Sum_{k>0} x^k/(k!-x^k). - _Vladeta Jovovic_, Oct 13 2003
%F A061095 a(n) ~ n!. - _Vaclav Kotesovec_, Jul 21 2019
%e A061095 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.
%p A061095 A061095 := n -> add(n!/(n/d)!^d, d = numtheory[divisors](n));
%p A061095 seq(A061095 (n), n = 1..17); # _Peter Luschny_, Apr 13 2011
%t A061095 Table[Sum[n!/(n/d)!^d,{d,Divisors[n]}],{n,1,20}]  (* _Geoffrey Critzer_, Aug 18 2011 *)
%o A061095 (PARI) /* compare to A038041 */
%o A061095 mnom(v)=
%o A061095 /* Multinomial coefficient s! / prod(j=1,n, v[j]!) where
%o A061095   s= sum(j=1,n, v[j]) and n is the number of elements in v[]. */
%o A061095 sum(j=1,#v, v[j])! / prod(j=1,#v, v[j]!)
%o A061095 A061095(n)={local(r=0);fordiv(n,d,r+=mnom(vector(d,j,n/d)));return(r);}
%o A061095 vector(33,n,A061095(n)) /* _Joerg Arndt_, Apr 16 2011 */
%o A061095 (PARI) a(n)=sumdiv(n,d, n!/(n/d)!^d ); \\ _Joerg Arndt_, Feb 23 2014
%Y A061095 Cf. A038041, A060538.
%Y A061095 Column k=1 of A327803.
%K A061095 nonn,nice,easy
%O A061095 1,2
%A A061095 _Henry Bottomley_, May 29 2001