cp's OEIS Frontend

A097088 Numerators of coefficients in function A(x) such that A(A(x)) = x+x^2.

%I A097088 #31 Jun 10 2018 04:24:47
%S A097088 0,1,1,-1,1,-5,27,-9,171,-69,-579,10689,-60261,116535,-304555,268707,
%T A097088 7942071,-19570935,9537731,1117836325,-630737297,-52310180977,
%U A097088 618435378229,523526983623,-3672122551119,8661572895987,1205887924659627,-8604836834766111,-77855893119175779
%N A097088 Numerators of coefficients in function A(x) such that A(A(x)) = x+x^2.
%C A097088 A097089 lists the exponents of 2 that form the reduced denominators.
%H A097088 Dmitry Kruchinin, Vladimir Kruchinin, <a href="https://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.
%F A097088 G.f.: A(x) = Sum_{n>=0} a(n)/2^A097089(n) where A(A(x)) = x + x^2.
%F A097088 a(n) = numerator(T(n,1)) if n>0, a(0) = 1, where T(n,m) = 1 if n=m, else T(n,m) = C(m,n-m) - Sum_{i=m+1..n-1} T(n,i)*T(i,m)/2. - _Vladimir Kruchinin_, Nov 08 2011
%t A097088 T[n_, m_] := T[n, m] = If[n == m, 1, Binomial[m, n-m] - Sum[T[n, i]*T[i, m]/2, {i, m+1, n-1}]/2]; a[n_] := T[n, 1] // Numerator; Table[a[n], {n, 0, 28}] (* _Jean-François Alcover_, Oct 29 2015, after _Vladimir Kruchinin_ *)
%o A097088 (PARI) {a(n)=local(A,B,F=x+x^2+x*O(x^n));A=F; if(n==0,0, for(i=0,n, B=serreverse(A); A=(A+subst(B,x,F))/2); numerator(polcoeff(A,n,x)))}
%o A097088 (Maxima) T(n,m):= if n=m then 1 else (binomial(m,n-m) -sum(T(n,i) *T(i,m), i,m+1,n-1))/2; makelist(num(T(n,1)), n,1,7); /* _Vladimir Kruchinin_, Nov 08 2011 */
%Y A097088 Cf. A097089.
%K A097088 sign,frac
%O A097088 0,6
%A A097088 _Paul D. Hanna_, Jul 23 2004