A244164 - OEIS

%I #22 Nov 27 2019 07:54:10

%S 1,1,3,6,15,23,53,94,203,404,855,1648,3416,6662,13400,26406,53038,

%T 105306,212051,422162,849267,1696864,3406077,6807024,13642099,

%U 27268122,54576003,109096436,218250874,436243705,872533347,1744312748,3488432736,6974783481

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

%H 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 Vaclav Kotesovec, <a href="/A244164/a244164_1.jpg">Graph a(n)/2^n</a>

%F a(n) = 2^(n-1) - A240085(n). - _Gus Wiseman_, Nov 25 2019

%e From _Gus Wiseman_, Nov 25 2019: (Start)

%e The a(1) = 1 through a(5) = 15 compositions:

%e   (1)  (2)  (3)    (4)      (5)

%e             (1,2)  (1,3)    (1,4)

%e             (2,1)  (3,1)    (2,3)

%e                    (1,1,2)  (3,2)

%e                    (1,2,1)  (4,1)

%e                    (2,1,1)  (1,1,3)

%e                             (1,2,2)

%e                             (1,3,1)

%e                             (2,1,2)

%e                             (2,2,1)

%e                             (3,1,1)

%e                             (1,1,1,2)

%e                             (1,1,2,1)

%e                             (1,2,1,1)

%e                             (2,1,1,1)

%e (End)

%p b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,

%p       add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))

%p     end:

%p a:= n-> b(n$2, 0, 1) -b(n$2, 0, 2):

%p seq(a(n), n=1..50);

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Min@@Length/@Split[Sort[#]]==1&]],{n,0,10}] (* _Gus Wiseman_, Nov 25 2019 *)

%Y Column k=1 of A242451.

%Y The complement is counted by A240085.

%Y Cf. A003242, A098504, A114901, A261983, A329740, A329741.

%K nonn

%O 1,3

%A _Alois P. Heinz_, Jun 21 2014