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A216688
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Expansion of e.g.f. exp( x * exp(x^2) ).
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21
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1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767
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OFFSET
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0,4
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LINKS
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FORMULA
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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]
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)
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MATHEMATICA
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With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}],
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PROG
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(PARI)
x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x^2) )))
(PARI) a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022
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CROSSREFS
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Cf. A216507 (e.g.f. exp(x^2*exp(x)), A216689 (e.g.f. exp(x*exp(x)^2)).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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