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%I #27 Jun 24 2022 23:37:47
%S 1,2,11,180,5805,298486,22228975,2258856824,300194704049,
%T 50529463186170,10505093602625139,2643441560563225468,
%U 791779611505017309493,278371498870260182630654,113516551713466910954246903,53143864598655784249290736512,28309328562668956145157858372537
%N Partial sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.
%H Vincenzo Librandi, <a href="/A217391/b217391.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k=0..n} t(k)^2 where t = A000670 (ordered Bell numbers).
%F a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - _Vaclav Kotesovec_, Nov 08 2014
%t t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k]^2, {k, 0, n}], {n, 0, 100}]
%o (Maxima)
%o t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o makelist(sum(t(k)^2,k,0,n),n,0,40);
%o (Magma)
%o A000670:=func<n | &+[StirlingSecond(n,i)*Factorial(i): i in [0..n]]>;
%o [&+[A000670(k)^2: k in [0..n]]: n in [0..14]]; // _Bruno Berselli_, Oct 03 2012
%o (PARI) for(n=0,30, print1(sum(k=0,n, (sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ _G. C. Greubel_, Feb 07 2018
%Y Cf. A000670, A006957, A005649, A217388, A217389, A217392.
%Y Partial sums of A122725.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Oct 02 2012
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