login
Self-convolution of numbers of preferential arrangements.
5

%I #33 Jun 01 2024 11:51:20

%S 1,2,7,32,185,1310,11067,109148,1234045,15752858,224169407,3518636504,

%T 60381131265,1124390692886,22577494959427,486212633129300,

%U 11177317486573445,273173247028616594,7072436847620016327,193351544314753174736,5565941751233499986185

%N Self-convolution of numbers of preferential arrangements.

%H Vincenzo Librandi, <a href="/A006957/b006957.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 08 2014

%F G.f.: (Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k*x))^2. - _Ilya Gutkovskiy_, Apr 06 2019

%p f:= proc(n) option remember; `if`(n<=1, 1,

%p add(binomial(n, k) *f(n-k), k=1..n))

%p end:

%p a:= n-> add(f(k)*f(n-k), k=0..n):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Feb 02 2009

%t t[n_] := Sum[StirlingS2[n, k]*k!, {k, 0, n}]; Table[Sum[t[k]*t[n-k], {k, 0, n}], {n, 0, 20}] (* _Jean-François Alcover_, Apr 09 2014, after _Emanuele Munarini_ *)

%o (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);

%o makelist(sum(t(k)*t(n-k),k,0,n),n,0,20); /* _Emanuele Munarini_, Oct 02 2012 */

%Y Cf. A000670, A217388, A217389.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Alois P. Heinz_, Feb 02 2009