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

%I A262046 #15 Nov 27 2017 11:08:19
%S A262046 0,0,2,6,54,460,3890,42364,512806,6698724,98496252,1585046584,
%T A262046 27568171818,520043947020,10550553510016,228796551051436,
%U A262046 5291441028244966,129967582592816500,3377869204044947060,92652519380506887784,2674716530794339146244
%N A262046 Number of ordered partitions of [n] such that at least two adjacent parts have the same size.
%C A262046 All terms are even.
%H A262046 Alois P. Heinz, <a href="/A262046/b262046.txt">Table of n, a(n) for n = 0..424</a>
%F A262046 a(n) ~ n! / (2 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Nov 27 2017
%p A262046 g:= proc(n) option remember; `if`(n<2, 1,
%p A262046        add(binomial(n, k)*g(k), k=0..n-1))
%p A262046     end:
%p A262046 b:= proc(n, i) option remember; `if`(n=0, 0, add(
%p A262046       `if`(i=j, g(n-j), b(n-j, j))*binomial(n, j), j=1..n))
%p A262046     end:
%p A262046 a:= n-> b(n, 0):
%p A262046 seq(a(n), n=0..25);
%t A262046 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 *)
%Y A262046 Cf. A000670, A261983, A262047.
%K A262046 nonn
%O A262046 0,3
%A A262046 _Alois P. Heinz_, Sep 09 2015