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 A140049 E.g.f. A(x) satisfies: A( x*exp(-x*A(x)) ) = exp(x*A(x)). 2
 1, 1, 5, 55, 1005, 26601, 941863, 42372177, 2336926665, 153927536545, 11869936146891, 1055015092106889, 106731589524249517, 12163935655214359329, 1548324822731892094191, 218516875165035215308801, 33979477899236956531288977, 5790103152487972170694748097 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..282 FORMULA a(n) = A140054(n+1)/(n+1). 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)) ). From Paul D. Hanna, Jul 09 2009: (Start) 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). ... 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) EXAMPLE 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! +... MAPLE 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): seq(a(n), n=0..20); # Alois P. Heinz, Aug 21 2019 MATHEMATICA m = 18; A[_] = 0; Do[A[x_] = Exp[x A[x] A[x A[x]]] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] * Range[0, m-1]! (* Jean-François Alcover, Oct 03 2019 *) PROG (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)} From Paul D. Hanna, Jul 09 2009: (Start) (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) CROSSREFS Cf. A140054, A140055. Cf. A162659. [From Paul D. Hanna, Jul 09 2009] Sequence in context: A203013 A266481 A006150 * A300589 A130031 A336289 Adjacent sequences: A140046 A140047 A140048 * A140050 A140051 A140052 KEYWORD nonn AUTHOR Paul D. Hanna, May 06 2008 STATUS approved

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Last modified February 5 12:39 EST 2023. Contains 360084 sequences. (Running on oeis4.)