A178533 E.g.f. satisfies: A(x) = Sum_{n>=0} A_{n}(x)^(n+1)/n! where A_{n}(x) denotes the n-th iteration of A(x).

1, 2, 15, 220, 5025, 159066, 6515551, 330344736, 20099812977, 1434653999650, 118051188070431, 11045624947772328, 1162186160312303737, 136250833163724567666, 17661781851438961602255, 2514858304152906320391136

Offset

1,2

Comments

Not the same as A140054.

Links

Table of n, a(n) for n=1..16.

Formula

E.g.f. satisfies: A_{k}(x) = Sum_{n>=0} A_{n+k-1}(x)^(n+1)/n! where A_{n}(x) denotes the n-th iteration of A(x): A_{n}(x) = A_{n-1}(A(x)) with A_0(x)=x.

Example

E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
The e.g.f. equals the series:
A(x) = x + A(x)^2 + A(A(x))^3/2! + A(A(A(x)))^4/3! + A(A(A(A(x))))^5/4! +...+ A_{n}(x)^(n+1)/n! +...
Related expansions:
A(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
A(A(A(x))) = x + 6*x^2/2! + 81*x^3/3! + 1776*x^4/4! + 55125*x^5/5! +...
A(A(A(A(x)))) = x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 121080*x^5/5! +...
The series reversion of the e.g.f. is given by:
A_{-1}(x) = x - x^2 - A(A(x))^3/2! - A(A(A(x)))^4/3! - A(A(A(A(x))))^5/4! -...- A_{n-1}(x)^(n+1)/n! -...
A_{-1}(x) = x - 2*x^2/2! - 3*x^3/3! - 40*x^4/4! - 795*x^5/5! - 22356*x^6/6! -...

Prog

(PARI) {ITERATE(F, n, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=sum(m=0, n, ITERATE(A, m, n)^(m+1)/m!)); n!*polcoeff(A, n)}

Crossrefs

Cf. A140054.

Sequence in context: A361617 A132493 A135860 * A087962 A140054 A099085

Adjacent sequences: A178530 A178531 A178532 * A178534 A178535 A178536

Keyword

nonn

Author

Paul D. Hanna, Dec 24 2010

Status

approved