The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A261982 Number of compositions of n with some part repeated. 54
 0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..3322 Wikipedia, Permutation pattern Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid. FORMULA a(n) = A011782(n) - A032020(n). G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020 EXAMPLE a(2) = 1: 11. a(3) = 1: 111. a(4) = 5: 22, 211, 121, 112, 1111. MAPLE b:= proc(n, k) option remember; `if`(k<0 or n<0, 0, `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1))) end: a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)): seq(a(n), n=0..40); MATHEMATICA b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *) Table[Length[Join@@Permutations/@Select[IntegerPartitions[n], Length[#]>Length[Split[#]]&]], {n, 0, 10}] (* Gus Wiseman, Jun 24 2020 *) CROSSREFS Row sums of A261981 and of A262191. Cf. A262047. The version for patterns is A019472. The (1,1)-avoiding version is A032020. The case of partitions is A047967. (1,1,1)-matching compositions are counted by A335455. Patterns matched by compositions are counted by A335456. (1,1)-matching compositions are ranked by A335488. Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755. Sequence in context: A168642 A357750 A234597 * A296033 A296968 A184552 Adjacent sequences: A261979 A261980 A261981 * A261983 A261984 A261985 KEYWORD nonn AUTHOR Alois P. Heinz, Sep 07 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 10 06:17 EDT 2024. Contains 375773 sequences. (Running on oeis4.)