A232685 G.f. A(x) = Sum_{n>=0} A_{2*n}(x)^(n+1) where A_{n+1}(x) = A_n(A(x)) with A_0(x)=x and A(0)=0.

1, 1, 5, 41, 433, 5361, 74261, 1120425, 18103665, 309717425, 5565231157, 104412465657, 2036253013681, 41135481319057, 858451844220693, 18465879790697993, 408688667270448497, 9292446000697905585, 216785984417061963573, 5183523630387054518873, 126912390721343458211761

Offset

1,3

Links

Table of n, a(n) for n=1..21.

Formula

G.f. satisfies: A_{k}(x) = Sum_{n>=0} A_{2*n+k}(x)^(n+1).

Example

 G.f.: A(x) = x + x^2 + 5*x^3 + 41*x^4 + 433*x^5 + 5361*x^6 + 74261*x^7 +...
where
A(x) = x + A(A(x))^2 + A(A(A(A(x))))^3 + A(A(A(A(A(A(x))))))^4 +...
A(A(x)) = A(x) + A(A(A(x)))^2 + A(A(A(A(A(x)))))^3 + A(A(A(A(A(A(A(x)))))))^4 +...
The series reversion of A(x) may be expressed by the series:
A_{-1}(x) = x - A(x)^2 - A(A(A(x)))^3 - A(A(A(A(A(x)))))^4 - A(A(A(A(A(A(A(x)))))))^5 -...
Explicitly,
A(A(x)) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1212*x^5 + 15696*x^6 + 225364*x^7 +...
A_{-1}(x) = x - x^2 - 3*x^3 - 21*x^4 - 203*x^5 - 2365*x^6 - 31275*x^7 - 454413*x^8 -...

Prog

 (PARI) {a(n)=local(A=x, G, S); if(n<1, 0, for(j=1, n, G=x; S=x; for(i=2, n, G=subst(A, x, subst(A, x, G+x*O(x^n))); S=S+G^i); A=S); polcoeff(A+x*O(x^n), n))}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A154757.

Sequence in context: A064087 A329123 A285064 * A081215 A218219 A140095

Adjacent sequences: A232682 A232683 A232684 * A232686 A232687 A232688

Keyword

nonn

Author

Paul D. Hanna, Nov 30 2013

Status

approved