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E.g.f. A(x) satisfies A(A(x)) = x/(1-x)^2.
3

%I #27 Jan 13 2024 10:52:06

%S 1,2,3,6,15,0,315,1890,-82215,708750,41008275,-1385549550,

%T -33403344975,3426898600125,26529571443375,-13516476003780750,

%U 157765729690193625,84230651703487038750,-3280917943856839411125,-799561865724400084556250,62859004972802312944044375

%N E.g.f. A(x) satisfies A(A(x)) = x/(1-x)^2.

%C First non-integer term is a(30) = 16103946844555056574100466078211185438823359375/2.

%H R. J. Mathar, <a href="/A091138/b091138.txt">Table of n, a(n) for n = 1..28</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

%F a(n) = n!* A030274(n)/A030275(n).

%F a(n) = n!*T(n,1), T(n,m)=1/2*(binomial(n+m-1,2*m-1)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 14 2012

%t t[n_, m_] := t[n, m] = If[n == m, 1, 1/2*(Binomial[n+m-1, 2*m-1] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}])]; a[n_] := n!*t[n, 1]; Table[a[n], {n, 1, 21}] (* _Jean-François Alcover_, Feb 26 2013, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o T(n,m):=if n=m then 1 else 1/2*(binomial(n+m-1,2*m-1)-sum(T(n,i)*T(i,m),i,m+1,n-1));

%o makelist(2^(n-1)*T(n,1),n,1,10); /* _Vladimir Kruchinin_, Mar 14 2012 */

%K easy,frac,fini,sign

%O 1,2

%A _Vladeta Jovovic_, Dec 20 2003

%E More terms from _R. J. Mathar_, Apr 28 2007