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%I A140054 #22 Oct 03 2019 10:02:05
%S A140054 1,2,15,220,5025,159606,6593041,338977416,21032339985,1539275365450,
%T A140054 130569297615801,12660181105282668,1387510663815243721,
%U A140054 170295099173001030606,23224872340978381412865,3496270002640563444940816,577651124287028261031912609,104221856744783499072505465746
%N A140054 E.g.f. A(x) satisfies: A( x*exp(-A(x)) ) = x.
%C A140054 Unsigned version of A087962.
%C A140054 Not the same as A178533.
%H A140054 Alois P. Heinz, <a href="/A140054/b140054.txt">Table of n, a(n) for n = 1..282</a> (first 75 terms from Paul D. Hanna)
%F A140054 E.g.f. A(x) satisfies:
%F A140054 (1) A(x) = x*exp( A(A(x)) ).
%F A140054 (2) A(x) = x*exp( A(x)*exp( A(A(x))*exp( A(A(A(x)))*exp( ...)))) (infinite exponential tower).
%F A140054 (3) Let A_{n}(x) denote n-th iteration of e.g.f. A(x) with A_0(x)=x,
%F A140054 then
%F A140054 (3.a) A_{n+1}(x) = A( A_{n}(x) ) = A_{n}(x) * exp( A_{n+2}(x) );
%F A140054 (3.b) A_{n}(x) = x*exp( Sum_{k=2..n+1} A_{k}(x) ).
%F A140054 (4) exp(-A(x)) = G(x) where G(x*G(x)) = exp(-x) and G(-x) = e.g.f. of A087961.
%F A140054 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]
%e A140054 E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 220*x^4/4! + 5025*x^5/5! +...
%e A140054 Related expansions are:
%e A140054 exp(-A(x)) = 1 - x - x^2/2! - 10*x^3/3! - 159*x^4/4! - 3816*x^5/5! -...
%e A140054 A(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 764*x^4/4! + 20400*x^5/5! +...
%e A140054 A(A(A(x))) = x + 6*x^2/2! + 81*x^3/3! + 1776*x^4/4! + 55125*x^5/5! +...
%e A140054 A(A(A(A(x)))) = x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 121080*x^5/5! +...
%e A140054 Iterations of A(x) obey the relation illustrated by:
%e A140054 A(x) = x*exp( A(A(x)) );
%e A140054 A(A(x)) = x*exp( A(A(x)) + A(A(A(x))) );
%e A140054 A(A(A(x))) = x*exp( A(A(x)) + A(A(A(x))) + A(A(A(A(x)))) ).
%e A140054 ...
%p A140054 b:= proc(n, k) option remember; `if`(n=0, 1/k, add(k*
%p A140054       b(j-1, j)*j*b(n-j, k)*binomial(n-1, j-1), j=1..n))
%p A140054     end:
%p A140054 a:= n-> b(n-1, n)*n:
%p A140054 seq(a(n), n=1..20);  # _Alois P. Heinz_, Aug 21 2019
%t A140054 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}]];
%t A140054 a[n_] := b[n - 1, n]*n;
%t A140054 a /@ Range[1, 20] (* _Jean-François Alcover_, Oct 03 2019, after _Alois P. Heinz_ *)
%o A140054 (PARI) {a(n)=local(A=x);for(i=0,n,A=serreverse(x*exp(-A+x*O(x^n))));n!*polcoeff(A,n)}
%o A140054 for(n=1,20,print1(a(n),", "))
%o A140054 (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)}
%o A140054 for(n=1,20,print1(a(n),", "))
%o A140054 (Maxima) A(n,m):=if n=m then 1 else m/n*sum(A(n-m,k)*n^k/k!,k,1,n-m);
%o A140054 makelist(n!*A(n,1),n,1,10); [_Vladimir Kruchinin_, May 06 2012]
%Y A140054 Cf. A087962 (A(-x)), A087961 (exp(-A(-x))), A140055 (A(A(x))).
%Y A140054 Cf. A178533.
%K A140054 nonn
%O A140054 1,2
%A A140054 _Paul D. Hanna_, May 03 2008

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