A155134 G.f. satisfies: A(x) = x + A( A(x)^2 + A(x)^3 ).

1, 1, 3, 11, 46, 207, 979, 4796, 24124, 123862, 646494, 3420055, 18296693, 98817934, 538065697, 2950537937, 16279888522, 90317357884, 503498509274, 2819104999847, 15846101219100, 89386208065828, 505843586002176

Offset

1,3

Links

Table of n, a(n) for n=1..23.

Formula

G.f. satisfies: A( x - A(x^2 + x^3) ) = x.

Given g.f. A(x), let G(x) = A(x)^2 + A(x)^3, then: A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +...

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 207*x^6 + 979*x^7 +...
A(A(x)^2 + A(x)^3) = x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 207*x^6 + 979*x^7 +...
Series_Reversion(A(x)) = x - A(x^2 + x^3) where
A(x^2 + x^3) = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 9*x^7 + 20*x^8 + 47*x^9 + 112*x^10 +...
Let G(x) = A(x)^2 + A(x)^3, then iterations of G(x) begin:
G(x) = x^2 + 3*x^3 + 10*x^4 + 40*x^5 + 175*x^6 + 812*x^7 + 3924*x^8 +...;
G(G(x)) = x^4 + 6*x^5 + 32*x^6 + 167*x^7 + 871*x^8 + 4575*x^9 +...;
G(G(G(x))) = x^8 + 12*x^9 + 100*x^10 + 718*x^11 + 4773*x^12 +...;
G(G(G(G(x)))) = x^16 + 24*x^17 + 344*x^18 + 3836*x^19 +...; ...
where A(x) = x + G(x) + G(G(x)) + G(G(G(x))) +...

Prog

(PARI) {a(n)=local(A=x+x*O(x^n)); if(n<1, 0, for(i=0, ceil(log(n+1)/log(2)), A=serreverse(x-subst(A, x, x^2+x^3+x^2*O(x^n)))); polcoeff(A, n))}

Crossrefs

Sequence in context: A151141 A275757 A291090 * A006605 A357233 A306822

Adjacent sequences: A155131 A155132 A155133 * A155135 A155136 A155137

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2009

Status

approved