Search: a140055 -id:a140055
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A140049
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E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)).
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1, 1, 5, 55, 1005, 26601, 941863, 42372177, 2336926665, 153927536545, 11869936146891, 1055015092106889, 106731589524249517, 12163935655214359329, 1548324822731892094191, 218516875165035215308801, 33979477899236956531288977, 5790103152487972170694748097
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f.: A(x) = exp(G(x)) where G(x) = e.g.f. of A140055.
E.g.f. satisfies: A(x) = exp( x*A(x) * A(x*A(x)) ).
E.g.f. satisfies: A(x) = exp(x*A(x)*A(x*A(x))).
...
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then
a(n,m) = Sum_{k=0..n} C(n,k) * m*(n+m)^(k-1) * a(n-k,k).
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Let log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then
L(n) = Sum_{k=1..n} C(n,k) * n^(k-1) * a(n-k,k).
(End)
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EXAMPLE
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A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
Log(A(x)) = G(x) = e.g.f. of A140055:
Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
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MAPLE
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b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*j
*b(j-1, j)*b(n-j, k)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, n+1):
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MATHEMATICA
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m = 18; A[_] = 0;
Do[A[x_] = Exp[x A[x] A[x A[x]]] + O[x]^m // Normal, {m}];
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PROG
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(PARI) {a(n)=local(A=x); for(i=0, n, A=serreverse(x*exp(-A+x*O(x^n)))); n!*polcoeff(A, n+1)}
(PARI) {a(n)=local(A=x); for(i=0, n, A=x*exp(subst(A, x, A+x*O(x^n)))); n!*polcoeff(A, n+1)}
(PARI) {a(n, m=1)=if(n==0, 1, if(m==0, 0^n, sum(k=0, n, binomial(n, k)*m*(n+m)^(k-1)*a(n-k, k))))}
(PARI) /* Log(A(x)) = x*A(x*A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */
{L(n)=if(n<1, 0, sum(k=1, n, binomial(n, k)*n^(k-1)*a(n-k, k)))} (End)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A140054
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E.g.f. A(x) satisfies: A( x*exp(-A(x)) ) = x.
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4
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1, 2, 15, 220, 5025, 159606, 6593041, 338977416, 21032339985, 1539275365450, 130569297615801, 12660181105282668, 1387510663815243721, 170295099173001030606, 23224872340978381412865, 3496270002640563444940816, 577651124287028261031912609, 104221856744783499072505465746
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) A(x) = x*exp( A(A(x)) ).
(2) A(x) = x*exp( A(x)*exp( A(A(x))*exp( A(A(A(x)))*exp( ...)))) (infinite exponential tower).
(3) Let A_{n}(x) denote n-th iteration of e.g.f. A(x) with A_0(x)=x,
then
(3.a) A_{n+1}(x) = A( A_{n}(x) ) = A_{n}(x) * exp( A_{n+2}(x) );
(3.b) A_{n}(x) = x*exp( Sum_{k=2..n+1} A_{k}(x) ).
(4) exp(-A(x)) = G(x) where G(x*G(x)) = exp(-x) and G(-x) = e.g.f. of A087961.
a(n)=n!*T(n,1), T(n,m)=m/n*sum(T(n-m,k)*n^k/k!,k,1,n-m), n>m, T(n.n)=1. [Vladimir Kruchinin, May 06 2012]
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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
Related expansions are:
exp(-A(x)) = 1 - x - x^2/2! - 10*x^3/3! - 159*x^4/4! - 3816*x^5/5! -...
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! +...
Iterations of A(x) obey the relation illustrated by:
A(x) = x*exp( A(A(x)) );
A(A(x)) = x*exp( A(A(x)) + A(A(A(x))) );
A(A(A(x))) = x*exp( A(A(x)) + A(A(A(x))) + A(A(A(A(x)))) ).
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MAPLE
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b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n-1, n)*n:
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MATHEMATICA
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b[n_, k_] := b[n, k] = If[n == 0, 1/k, Sum[k*b[j - 1, j]*j*b[n - j, k]* Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n - 1, n]*n;
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PROG
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(PARI) {a(n)=local(A=x); for(i=0, n, A=serreverse(x*exp(-A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=x); for(i=0, n, A=x*exp(subst(A, x, A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(Maxima) A(n, m):=if n=m then 1 else m/n*sum(A(n-m, k)*n^k/k!, k, 1, n-m);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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