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%I #11 Nov 27 2017 11:51:52
%S 1,8,281,21328,2858481,596558808,179058197641,73110755339168,
%T 38977936014004961,26295624802015360168,21898514473870334203641,
%U 22064773395630274673891568,26456951179676525013504937681,37229662306608638451691410580088
%N Alternating sums of the squares of the numbers in sequence A080253
%F a(n) = sum((-1)^(n-k)*c(k)^2,k=0..n), where c(n) = A080253(n).
%F a(n) ~ (n!)^2 * 2^(2*n-1) / (log(2))^(2*n + 2). - _Vaclav Kotesovec_, Nov 27 2017
%t t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[(-1)^(n-k)c[k]^2, {k,0,n}], {n,0,100}]
%o (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);
%o c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
%o makelist(sum((-1)^(n-k)*c(k)^2,k,0,n),n,0,40);
%Y Cf. A080253, A000670, A217483, A217484, A217485, A217486, A217487.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Oct 04 2012