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

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

Offset: 0

Author

Alois P. Heinz, Sep 09 2015

Keywords

• nonn

Comments

All terms are even.

Links

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

Crossrefs

Cf. A000670, A261983, A262047.

Programs

Maple

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);

Mathematica

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 *)

Formula

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