A107700 G.f. A(x) satisfies: A(A(x)) = x + 2*A(x)^2.

1, 1, 1, 0, -2, 0, 13, 0, -145, 0, 2328, 0, -49784, 0, 1358965, 0, -46076544, 0, 1902202515, 0, -94104681660, 0, 5503867176832, 0, -376096374571125, 0, 29714871818774044, 0, -2689473418781240320, 0, 276562260699626541509, 0, -32073434441440654231749, 0

Offset: 1

Author

Paul D. Hanna, May 21 2005

Keywords

• sign

Comments

Inspired by peculiar functional equations suggested by Michael Somos. Unexpectedly, the even-indexed terms are all zeros after index 2; see A107699 for odd-indexed terms. The self-COMPOSE equals A107701.

Links

• P. Majer, The functional equation f(f(x))=x+f(x)^2, Dec. 2010

Crossrefs

Cf. A107699, A107701.

Programs

Mathematica

T[n_, n_] = 1; T[n_, m_] := T[n, m] = Sum[Binomial[m, j]*2^(m-j-1)*T[n-j, 2*(m-j)], {j, Max[2*m-n, 0], m-1}] - 1/2*Sum[T[n, i]*T[i, m], {i, m+1, n-1}]; Table[T[n, 1], {n, 1, 34}] (* Jean-François Alcover, Mar 03 2014, after Vladimir Kruchinin *)

Maxima

T(n,m):=if n=m then 1 else sum(binomial(m,j)*2^(m-j-1)*T(n-j,2*(m-j)),j,max(2*m-n,0),m-1)-1/2*sum(T(n,i)*T(i,m),i,m+1,n-1);
makelist(T(n,1),n,1,9); /* Vladimir Kruchinin, Mar 12 2012 */

PARI

{a(n) = local(A,B,F); A=x+x^2+x*O(x^n); if(n<1, 0, for(i=0, n, F=x+2*A^2; B=serreverse(A); A=(A+subst(B,x,F))/2); polcoeff(A,n,x))}

PARI

{a(n) = my(A); if( n<0, 0, A = x + O(x^2); for(k=2, n, A = truncate(A) + x * O(x^k); A += (x + A^2 - subst(A, x, A))/2); polcoeff(A, n) * 2^(n-1))}; /* Michael Somos, Dec 15 2017 */

Formula

G.f. satisfies: A(-A(-x)) = x.

G.f. satisfies: A( A(x) - 2*x^2 ) = x. [Paul D. Hanna, Aug 20 2008]

a(n)=T(n,1), T(n,m)=sum(j=max(2*m-n,0)..m-1,binomial(m,j)*2^(m-j-1) *T(n-j,2*(m-j)))-1/2*sum(i=m+1..n-1, T(n,i)*T(i,m)), n>m, T(n,n)=1. [Vladimir Kruchinin, Mar 12 2012]