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

0,6

Comments

A097089 lists the exponents of 2 that form the reduced denominators.

Links

Table of n, a(n) for n=0..28.

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

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

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 *)

Prog

(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)))}
(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 */

Crossrefs

Cf. A097089.

Sequence in context: A342997 A118391 A196085 * A196009 A091721 A039283

Adjacent sequences: A097085 A097086 A097087 * A097089 A097090 A097091

Keyword

sign,frac

Author

Paul D. Hanna, Jul 23 2004

Status

approved