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
Keywords
Examples
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! -...
Crossrefs
Cf. A140054.
Programs
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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)}
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.
Comments