A262047 -id:A262047 - cp's OEIS frontend

A261982 Number of compositions of n with some part repeated.

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

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

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.

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.

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);

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 *)

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

A262046 Number of ordered partitions of [n] such that at least two adjacent parts have the same size.

Original entry on oeis.org

0, 0, 2, 6, 54, 460, 3890, 42364, 512806, 6698724, 98496252, 1585046584, 27568171818, 520043947020, 10550553510016, 228796551051436, 5291441028244966, 129967582592816500, 3377869204044947060, 92652519380506887784, 2674716530794339146244

Offset: 0

Alois P. Heinz, Sep 09 2015

nonn

All terms are even.

Alois P. Heinz, Table of n, a(n) for n = 0..424

Cf. A000670, A261983, A262047.

g:= proc(n) option remember; `if`(n<2, 1,
       add(binomial(n, k)*g(k), k=0..n-1))
    end:
b:= proc(n, i) option remember; `if`(n=0, 0, add(
      `if`(i=j, g(n-j), b(n-j, j))*binomial(n, j), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);

g[n_] := g[n] = If[n<2, 1, Sum[Binomial[n, k]*g[k], {k, 0, n-1}]]; b[n_, i_] := b[n, i] = If[n==0, 0, Sum[If[i==j, g[n-j], b[n-j, j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, Nov 27 2017

cp's OEIS Frontend

A261982 Number of compositions of n with some part repeated.

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