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A006957 Self-convolution of numbers of preferential arrangements.

%I A006957 #44 Dec 27 2024 13:35:45
%S A006957 1,2,7,32,185,1310,11067,109148,1234045,15752858,224169407,3518636504,
%T A006957 60381131265,1124390692886,22577494959427,486212633129300,
%U A006957 11177317486573445,273173247028616594,7072436847620016327,193351544314753174736,5565941751233499986185
%N A006957 Self-convolution of numbers of preferential arrangements.
%H A006957 Vincenzo Librandi, <a href="/A006957/b006957.txt">Table of n, a(n) for n = 0..200</a>
%F A006957 a(n) ~ n! / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 08 2014
%F A006957 G.f.: (Sum_{j>=0} j!*x^j / Product_{k=1..j} (1 - k*x))^2. - _Ilya Gutkovskiy_, Apr 06 2019
%F A006957 a(n) = (n+1)! [x^(n+1)] (x-2*log(2-exp(x)))/(4-exp(x)). - _Ira M. Gessel_, Dec 26 2024
%p A006957 f:= proc(n) option remember; `if`(n<=1, 1,
%p A006957       add(binomial(n, k) *f(n-k), k=1..n))
%p A006957     end:
%p A006957 a:= n-> add(f(k)*f(n-k), k=0..n):
%p A006957 seq(a(n), n=0..25);  # _Alois P. Heinz_, Feb 02 2009
%t A006957 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 A006957 (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o A006957 makelist(sum(t(k)*t(n-k),k,0,n),n,0,20); /* _Emanuele Munarini_, Oct 02 2012 */
%o A006957 (PARI) a006957(n)=my(x='x+O('x^(n+2))); Vec((x-2*log(2-exp(x)))/(4-exp(x)))[n+1]*(n+1)! \\ _Hugo Pfoertner_, Dec 27 2024
%Y A006957 Cf. A000670, A217388, A217389.
%K A006957 nonn
%O A006957 0,2
%A A006957 _N. J. A. Sloane_
%E A006957 More terms from _Alois P. Heinz_, Feb 02 2009