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%I #18 Mar 01 2019 03:00:36
%S 1,0,0,6,0,0,360,0,0,60480,0,0,19958400,0,0,10897286400,0,0,
%T 8892185702400,0,0,10137091700736000,0,0,15388105201717248000,0,0,
%U 30006805143348633600000,0,0,73096577329197271449600000
%N Number of sets of lists (sequences) of n labeled elements with k=3 elements per list.
%C The (labeled) case for k=2 is A067994, the Hermite numbers. The (labeled) case for k>=1 is A000262, Number of "sets of lists".
%H Alois P. Heinz, <a href="/A101109/b101109.txt">Table of n, a(n) for n = 0..582</a>
%F E.g.f.: exp(z^3).
%F a(0) = 1, a(1) = 0, a(2) = 0, (-n-3)*a(n+3)+3*a(n).
%F a(n) = n!/(n/3)!, if 3 divides n, 0 otherwise. - _Mitch Harris_, Jan 19 2006
%e Let Z[i] denote the i-th labeled element. Then a(3) = 6 with the following six sets:
%e Set(Sequence(Z[3],Z[1],Z[2])), Set(Sequence(Z[2],Z[1],Z[3])), Set(Sequence(Z[3],Z[2],Z[1])), Set(Sequence(Z[2],Z[3],Z[1])), Set(Sequence(Z[1],Z[3],Z[2])), Set(Sequence(Z[1],Z[2],Z[3])).
%p A101109 := n -> n!*PIECEWISE([1/GAMMA(1/3*n+1), irem(n,3) = 0],[0, irem(n-1,3) = 0],[0, irem(n-2,3) = 0]); [ seq(n!*PIECEWISE([1/GAMMA(1/3*n+1), irem(n,3) = 0],[0, irem(n-1,3) = 0],[0, irem(n-2,3) = 0]),n=0..30) ];
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p j!*binomial(n-1, j-1), j=`if`(n>2, 3, [][])))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, May 10 2016
%t With[{nn=30},CoefficientList[Series[Exp[x^3],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 16 2013 *)
%o (Sage)
%o def A101109(n) : return factorial(n)/factorial(n/3) if n%3 == 0 else 0
%o [A101109(n) for n in (0..30)] # _Peter Luschny_, Jul 12 2012
%Y Cf. A000262, A067994.
%K nonn
%O 0,4
%A _Thomas Wieder_, Dec 01 2004
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