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A220110 Expansion of A(x) satisfying A(A(x)) = x+2x^2+4x^3.

%I A220110 #25 May 03 2024 11:14:47
%S A220110 1,1,1,-3,5,1,-39,117,13,-1311,3441,9525,-78603,16961,1520521,
%T A220110 -3649323,-28760163,144787265,601582689,-5374096875,-15170850555,
%U A220110 225456060897,461284881657,-11141961064971,-15963771799251,647040052660257,569313149887057
%N A220110 Expansion of A(x) satisfying A(A(x)) = x+2x^2+4x^3.
%H A220110 Seiichi Manyama, <a href="/A220110/b220110.txt">Table of n, a(n) for n = 1..552</a>
%H A220110 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 A220110 a(n)=T(n,1), 2*T(n,m)= 2^(n-m) *sum_{j=0..m} binomial(j,n-3*m+2*j) *binomial(m,j) -sum_{k=m+1..n-1} T(n,k)*T(k,m), n>m, T(n,n)=1.
%e A220110 First column of
%e A220110 1;
%e A220110 1,1;
%e A220110 1,2,1;
%e A220110 -3,3,3,1;
%e A220110 5,-4,6,4,1;
%e A220110 1,5,-2,10,5,1;
%e A220110 -39,6,3,4,15,6,1;
%e A220110 117,-57,9,3,15,21,7,1;
%e A220110 13,128,-56,8,10,32,28,8,1;
%e A220110 -1311,201,84,-44,6,30,56,36,9,1;
%t A220110 t[n_, m_] := t[n, m] = 1/2*(2^(n-m)*Sum[Binomial[j, n - 3*m + 2*j]*Binomial[m, j], {j, 0, m}] - Sum[t[n, k]*t[k, m], {k, m+1, n-1}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 27}] (* _Jean-François Alcover_, Feb 22 2013 *)
%o A220110 (Maxima)
%o A220110 T(n,m):=if n=m then 1 else 1/2*(2^(n-m)*sum(binomial(j,n-3*m+2*j)*binomial(m,j),j,0,m)-sum(T(n,k)*T(k,m),k,m+1,n-1));
%o A220110 makelist((T(n,1)),n,1,10);
%K A220110 sign
%O A220110 1,4
%A A220110 _Dmitry Kruchinin_, Dec 05 2012