cp's OEIS Frontend

A217392 Alternating sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.

%I A217392 #19 Sep 08 2022 08:46:04
%S A217392 1,0,9,160,5465,287216,21643273,2214984576,295720862649,
%T A217392 49933547619472,10404630591819497,2622531836368780832,
%U A217392 786513638108085303193,276793205620647080017968,112961387008976003691598281,52917386659933341334644891328,28203267311410367019573922744697
%N A217392 Alternating sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.
%H A217392 Vincenzo Librandi, <a href="/A217392/b217392.txt">Table of n, a(n) for n = 0..200</a>
%F A217392 a(n) = sum((-1)^(n-k)*t(k)^2, k=0..n), where t = A000670 (ordered Bell numbers).
%F A217392 a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - _Vaclav Kotesovec_, Nov 08 2014
%t A217392 t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n-k)t[k]^2, {k, 0, n}], {n, 0, 100}]
%o A217392 (Maxima)
%o A217392 t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o A217392 makelist(sum((-1)^(n-k)*t(k)^2,k,0,n),n,0,40);
%o A217392 (Magma)
%o A217392 A000670:=func<n | &+[StirlingSecond(n,i)*Factorial(i): i in [0..n]]>;
%o A217392 [&+[(-1)^(n-k)*A000670(k)^2: k in [0..n]]: n in [0..14]]; // _Bruno Berselli_, Oct 03 2012
%o A217392 (PARI) for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*(sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ _G. C. Greubel_, Feb 07 2018
%Y A217392 Cf. A000670, A006957, A005649, A217388, A217389, A217391.
%K A217392 nonn
%O A217392 0,3
%A A217392 _Emanuele Munarini_, Oct 02 2012