A140054 E.g.f. A(x) satisfies: A( x*exp(-A(x)) ) = x.
1, 2, 15, 220, 5025, 159606, 6593041, 338977416, 21032339985, 1539275365450, 130569297615801, 12660181105282668, 1387510663815243721, 170295099173001030606, 23224872340978381412865, 3496270002640563444940816, 577651124287028261031912609, 104221856744783499072505465746
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! +... 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)))) ). ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..282 (first 75 terms from Paul D. Hanna)
Programs
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Maple
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: seq(a(n), n=1..20); # Alois P. Heinz, Aug 21 2019 -
Mathematica
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; a /@ Range[1, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *) -
Maxima
A(n,m):=if n=m then 1 else m/n*sum(A(n-m,k)*n^k/k!,k,1,n-m); makelist(n!*A(n,1),n,1,10); [Vladimir Kruchinin, May 06 2012]
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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),", "))
Formula
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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