A216689 - OEIS

%I #32 Oct 31 2024 08:55:49

%S 1,1,5,25,153,1121,9373,87417,898033,10052353,121492341,1573957529,

%T 21729801481,318121178337,4917743697805,79981695655801,

%U 1364227940101857,24335561350365953,452874096174214117,8772713803852981785,176541611843378273401,3684142819311127955041,79596388271096140589949

%N Expansion of e.g.f. exp( x * exp(x)^2 ).

%H Vincenzo Librandi, <a href="/A216689/b216689.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>

%F O.g.f.: Sum_{n>=0} x^n / (1 - 2*n*x)^(n+1). - _Paul D. Hanna_, Aug 02 2014

%F a(n) = Sum_{k=0..n} binomial(n,k) * (2*k)^(n-k) for n>=0. - _Paul D. Hanna_, Aug 02 2014

%F From _Vaclav Kotesovec_, Aug 06 2014: (Start)

%F a(n) ~ n^n / (exp(2*n*r/(1+2*r)) * r^n * sqrt((1+6*r+4*r^2)/(1+2*r))), where r is the root of the equation r*(1+2*r)*exp(2*r) = n.

%F (a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).

%F (End)

%t With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x]^2], {x, 0, nn}], x] Range[0, nn]!] (* _Bruno Berselli_, Sep 14 2012 *)

%o (PARI)

%o x='x+O('x^66);

%o Vec(serlaplace(exp( x * exp(x)^2 )))

%o /* _Joerg Arndt_, Sep 14 2012 */

%o (PARI) /* From o.g.f.: */

%o {a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - 2*k*x +x*O(x^n))^(k+1));polcoeff(A, n)}

%o for(n=0,25,print1(a(n),", ")) /* _Paul D. Hanna_, Aug 02 2014 */

%o (PARI) /* From binomial sum: */

%o {a(n)=sum(k=0,n, binomial(n,k)*(2*k)^(n-k))}

%o for(n=0,30,print1(a(n),", ")) /* _Paul D. Hanna_, Aug 02 2014 */

%Y Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216688 (e.g.f. exp(x*exp(x^2))).

%Y Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

%Y Cf. A240165 (e.g.f. exp(x*(1+exp(x)^2))).

%K nonn

%O 0,3

%A _Joerg Arndt_, Sep 14 2012