A097088 Numerators of coefficients in function A(x) such that A(A(x)) = x+x^2.
0, 1, 1, -1, 1, -5, 27, -9, 171, -69, -579, 10689, -60261, 116535, -304555, 268707, 7942071, -19570935, 9537731, 1117836325, -630737297, -52310180977, 618435378229, 523526983623, -3672122551119, 8661572895987, 1205887924659627, -8604836834766111, -77855893119175779
Offset: 0
Links
- Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
Crossrefs
Cf. A097089.
Programs
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Mathematica
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 *) -
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 */
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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)))}
Formula
G.f.: A(x) = Sum_{n>=0} a(n)/2^A097089(n) where A(A(x)) = x + x^2.
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
Comments