cp's OEIS Frontend

A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.

%I A244164 #22 Nov 27 2019 07:54:10
%S A244164 1,1,3,6,15,23,53,94,203,404,855,1648,3416,6662,13400,26406,53038,
%T A244164 105306,212051,422162,849267,1696864,3406077,6807024,13642099,
%U A244164 27268122,54576003,109096436,218250874,436243705,872533347,1744312748,3488432736,6974783481
%N A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.
%H A244164 Alois P. Heinz and Vaclav Kotesovec, <a href="/A244164/b244164.txt">Table of n, a(n) for n = 1..2000</a> (first 400 terms from Alois P. Heinz)
%H A244164 Vaclav Kotesovec, <a href="/A244164/a244164_1.jpg">Graph a(n)/2^n</a>
%F A244164 a(n) = 2^(n-1) - A240085(n). - _Gus Wiseman_, Nov 25 2019
%e A244164 From _Gus Wiseman_, Nov 25 2019: (Start)
%e A244164 The a(1) = 1 through a(5) = 15 compositions:
%e A244164   (1)  (2)  (3)    (4)      (5)
%e A244164             (1,2)  (1,3)    (1,4)
%e A244164             (2,1)  (3,1)    (2,3)
%e A244164                    (1,1,2)  (3,2)
%e A244164                    (1,2,1)  (4,1)
%e A244164                    (2,1,1)  (1,1,3)
%e A244164                             (1,2,2)
%e A244164                             (1,3,1)
%e A244164                             (2,1,2)
%e A244164                             (2,2,1)
%e A244164                             (3,1,1)
%e A244164                             (1,1,1,2)
%e A244164                             (1,1,2,1)
%e A244164                             (1,2,1,1)
%e A244164                             (2,1,1,1)
%e A244164 (End)
%p A244164 b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A244164       add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
%p A244164     end:
%p A244164 a:= n-> b(n$2, 0, 1) -b(n$2, 0, 2):
%p A244164 seq(a(n), n=1..50);
%t A244164 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Min@@Length/@Split[Sort[#]]==1&]],{n,0,10}] (* _Gus Wiseman_, Nov 25 2019 *)
%Y A244164 Column k=1 of A242451.
%Y A244164 The complement is counted by A240085.
%Y A244164 Cf. A003242, A098504, A114901, A261983, A329740, A329741.
%K A244164 nonn
%O A244164 1,3
%A A244164 _Alois P. Heinz_, Jun 21 2014