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 A161632 E.g.f. satisfies: A(x) = (1 + x*exp(x*A(x)))^2. 0
 1, 2, 6, 42, 392, 4970, 78492, 1489838, 33105648, 842437170, 24181696820, 772887702422, 27228973364232, 1048392980781770, 43802436902618604, 1973819502540516990, 95426799849067842272, 4927195390491532227170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA a(n) = n!*Sum_{k=0..n} C(2*(n-k+1),k)/(n-k+1) * k^(n-k)/(n-k)!. If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n!*Sum_{k=0..n} m*C(2*(n-k+m),k)/(n-k+m) * k^(n-k)/(n-k)!. E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = (1 + x*exp(x)/B(x))^2. a(n) ~ sqrt(2*s^(3/2)*(2-5*sqrt(s)+3*s)/(2*sqrt(s)-1)) * (2*s-2*sqrt(s))^n * n^(n-1) / exp(n), where s = 3.533778497303240223520495... is the root of the equation (2-2/sqrt(s)) * log(2*(sqrt(s)-2*s+s^(3/2))) = 1. - Vaclav Kotesovec, Jan 10 2014 EXAMPLE E.g.f.: A(x) = 1 + 2*x + 6*x^2/2! + 42*x^3/3! + 392*x^4/4! + 4970*x^5/5! +... A(x)^(1/2) = 1 + x + 2*x^2/2! + 15*x^3/3! + 124*x^4/4! + 1565*x^5/5! +... MATHEMATICA Flatten[{1, Table[n!*Sum[Binomial[2*(n-k+1), k]/(n-k+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, m*binomial(2*(n-k+m), k)/(n-k+m)*k^(n-k)/(n-k)!)} CROSSREFS Cf. A161631. Sequence in context: A050862 A227250 A258969 * A115974 A179330 A066864 Adjacent sequences:  A161629 A161630 A161631 * A161633 A161634 A161635 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 18 2009 STATUS approved

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Last modified April 4 05:34 EDT 2020. Contains 333212 sequences. (Running on oeis4.)