|
|
A216688
|
|
Expansion of e.g.f. exp( x * exp(x^2) ).
|
|
23
|
|
|
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
|
|
|
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]
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}],
|
|
PROG
|
(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
|
|
CROSSREFS
|
Cf. A216507 (e.g.f. exp(x^2*exp(x)), A216689 (e.g.f. exp(x*exp(x)^2)).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|