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A161633
E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).
28
1, 1, 4, 27, 268, 3525, 57966, 1146061, 26500552, 702069129, 20974309210, 697754762001, 25584428686620, 1025230366195789, 44579963354153878, 2090676600895922565, 105191995364927688976, 5652501986238910061073, 323083811850594613809714, 19573120681427758058921881
OFFSET
0,3
LINKS
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*exp(x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
MATHEMATICA
Flatten[{1, Table[n!*Sum[Binomial[n+1, k]/(n+1) * k^(n-k)/(n-k)!, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
PROG
(PARI) a(n, m=1)=n!*sum(k=0, n, binomial(n+m, k)*m/(n+m)*k^(n-k)/(n-k)!)
CROSSREFS
Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.
Sequence in context: A331316 A353233 A265270 * A052871 A104653 A194787
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 18 2009
STATUS
approved