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 A161565 E.g.f. satisfies: A(x) = exp(x*exp(2*x*A(x))). 5
 1, 1, 5, 37, 417, 6201, 115393, 2583141, 67643201, 2029868785, 68699859201, 2589393498429, 107580709769569, 4885086832499433, 240716442970201409, 12793428673619226901, 729511897042502788737, 44427614415877495608801 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..367 FORMULA a(n) = Sum_{k=0..n} 2^(n-k) * C(n,k) * (n-k+1)^(k-1) * k^(n-k). a(n) ~ sqrt(LambertW(1/r)) * n^(n-1) / (2*exp(n)*r^(n+1)), where r = 0.256263163133653382... is the root of the equation 1/LambertW(1/r) = -log(2*r^2) - LambertW(1/r). - Vaclav Kotesovec, Feb 28 2014 EXAMPLE E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 417*x^4/4! +... log(A(x)) = x*B(x) where B(x) = exp(2*x*A(x)) = e.g.f. of A161566: B(x) = 1 + 2*x + 8*x^2/2! + 62*x^3/3! + 696*x^4/4! + 10362*x^5/5! +... MATHEMATICA Flatten[{1, Table[Sum[2^(n-k) * Binomial[n, k] * (n-k+1)^(k-1) * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 28 2014 *) PROG (PARI) {a(n)=sum(k=0, n, 2^(n-k)*binomial(n, k) * (n-k+1)^(k-1) * k^(n-k))} (PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(x*exp(2*x*A+O(x^n)))); n!*polcoeff(A, n, x)} CROSSREFS Cf. A161566. Sequence in context: A209671 A173796 A292873 * A235345 A318002 A323567 Adjacent sequences:  A161562 A161563 A161564 * A161566 A161567 A161568 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 14 2009 STATUS approved

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Last modified December 2 05:01 EST 2021. Contains 349437 sequences. (Running on oeis4.)