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 A216688 Expansion of e.g.f. exp( x * exp(x^2) ). 21
 1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function FORMULA a(n)=(n!*sum(m=floor((n+1)/2)..n, ((2*m-n)^(n-m))/((2*m-n)!*(n-m)!))). [Vladimir Kruchinin, Mar 09 2013] From Vaclav Kotesovec, Aug 06 2014: (Start) a(n) ~ n^n / (r^n * exp((2*r^2*n)/(1+2*r^2)) * sqrt(3+2*r^2 - 2/(1 + 2*r^2))), where r is the root of the equation r*exp(r^2)*(1+2*r^2) = n. (a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(2^(1/3)*n^(2/3)/3))) * sqrt(2/(3*LambertW(2^(1/3)*n^(2/3)/3))). (End) MATHEMATICA With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *) PROG (PARI) x='x+O('x^66); Vec(serlaplace(exp( x * exp(x^2) ))) /* Joerg Arndt, Sep 14 2012 */ (PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022 CROSSREFS Cf. A216507 (e.g.f. exp(x^2*exp(x)), A216689 (e.g.f. exp(x*exp(x)^2)). Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))). Sequence in context: A266810 A199893 A129791 * A141626 A138292 A138738 Adjacent sequences: A216685 A216686 A216687 * A216689 A216690 A216691 KEYWORD nonn AUTHOR Joerg Arndt, Sep 14 2012 STATUS approved

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Last modified May 28 01:34 EDT 2024. Contains 372900 sequences. (Running on oeis4.)