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A240085 Number of compositions of n in which no part is unique (every part appears at least twice).

%I A240085 #28 Jan 10 2025 01:16:54
%S A240085 1,0,1,1,2,1,9,11,34,53,108,169,400,680,1530,2984,6362,12498,25766,
%T A240085 50093,102126,199309,400288,788227,1581584,3135117,6286310,12532861,
%U A240085 25121292,50184582,100627207,201208477,403170900,806534560,1615151111,3231224804,6467909442
%N A240085 Number of compositions of n in which no part is unique (every part appears at least twice).
%D A240085 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009.
%H A240085 Alois P. Heinz and Vaclav Kotesovec, <a href="/A240085/b240085.txt">Table of n, a(n) for n = 0..2100</a> (first 500 terms from Alois P. Heinz)
%H A240085 Vaclav Kotesovec, <a href="/A240085/a240085.jpg">Graph a(n)/2^n</a>
%F A240085 G.f.: 1 + Sum_{m>1} C(m,x)/(1-x^m) where C(m,x) = x^m + Sum_{i=2..m-2} m! * x^m * C(m-i,x)/(i! * (m-i)! * (1 - x^(m-i))). - _John Tyler Rascoe_, Jan 09 2025
%e A240085 a(6) = 9 because we have: 3+3, 2+2+2, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 1+1+1+1+1+1.
%p A240085 b:= proc(n, i, t) option remember; `if`(n=0, t!, `if`(i<1, 0,
%p A240085       b(n, i-1, t) +add(b(n-i*j, i-1, t+j)/j!, j=2..n/i)))
%p A240085     end:
%p A240085 a:= n-> b(n$2, 0):
%p A240085 seq(a(n), n=0..40);  # _Alois P. Heinz_, Mar 31 2014
%t A240085 Table[Length[Level[Map[Permutations,Select[IntegerPartitions[n], Apply[And,Table[Count[#,#[[i]]]>1,{i,1,Length[#]}]]&]],{2}]],{n,0,20}]
%t A240085 (* Second program: *)
%t A240085 b[n_, i_, t_] := b[n, i, t] = If[n == 0, t!, If[i < 1, 0, b[n, i - 1, t] + Sum[b[n - i*j, i - 1, t + j]/j!, {j, 2, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y A240085 Cf. A007690.
%K A240085 nonn
%O A240085 0,5
%A A240085 _Geoffrey Critzer_, Mar 31 2014
%E A240085 More terms from _Alois P. Heinz_, Mar 31 2014