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%I #11 Aug 06 2019 16:03:05
%S 0,1,2,2,2,2,0,4,6,-58,100,1052,-5924,-21972,322020,332392,-21168682,
%T 29068598,1724404180,-7070346036,-172304798980,1290100381724,
%U 20728501384592,-247269172883976,-2936888518668676,53037176259027580,477640220538178184
%N G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).
%C A(x) is sometimes called a functional square root, or half-iterate of F(x).
%H Gottfried Helms, <a href="http://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm">Coefficients for fractional iterates exp(x)-1</a>.
%H Dmitry Kruchinin, Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$</a>, arXiv:1302.1986 [math.CO], 2013.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Functional_square_root">Functional square root</a>.
%t half[q_] := half[q] = Module[{h}, h[0] = 0; h[1] = 1; h[n_Integer] := h[n] = Module[{c}, c[m_Integer /; m < n] := h[m]; c[n] /. Solve[q[n] == Sum[k! c[k] BellY[n, k, Table[m! c[m], {m, n - k + 1}]], {k, n}]/n!, c[n]][[1]]]; h]; a[n_Integer] := a[n] = half[Function[k, k 2^(k-1)]][n]; Table[a[n], {n, 0, 26}]
%Y Cf. A001787, A027436, A184011.
%K sign
%O 0,3
%A _Vladimir Reshetnikov_, Aug 05 2019
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