cp's OEIS Frontend

A240165 E.g.f.: exp( x*(1 + exp(2*x)) ).

%I A240165 #20 Jul 17 2016 11:41:54
%S A240165 1,2,8,44,288,2192,18976,182912,1934848,22231808,275203584,3645178880,
%T A240165 51370694656,766634946560,12066538676224,199607631945728,
%U A240165 3459736006950912,62662715180515328,1183139425871331328,23237689444403511296,473852525131782946816,10014501808427774246912
%N A240165 E.g.f.: exp( x*(1 + exp(2*x)) ).
%H A240165 Vincenzo Librandi, <a href="/A240165/b240165.txt">Table of n, a(n) for n = 0..200</a>
%H A240165 Vaclav Kotesovec, <a href="http://oeis.org/A216688/a216688.pdf">Asymptotic solution of the equations using the Lambert W-function</a>
%F A240165 O.g.f.: Sum_{n>=0} x^n / (1 - (2*n+1)*x)^(n+1).
%F A240165 a(n) = Sum_{k=0..n} binomial(n,k) * (2*k+1)^(n-k) for n>=0.
%F A240165 From _Vaclav Kotesovec_, Aug 06 2014: (Start)
%F A240165 a(n) ~ exp((1+exp(2*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(2*r)*r*(1 + 6*r + 4*r^2))), where r is the root of the equation r*(1 + exp(2*r) + 2*r*exp(2*r)) = n.
%F A240165 (a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
%F A240165 (End)
%e A240165 E.g.f.: E(x) = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 288*x^4/4! + 2192*x^5/5! +...
%e A240165 where E(x) = exp(x) * exp(x*exp(2*x)).
%e A240165 O.g.f.: A(x) = 1 + 2*x + 8*x^2 + 44*x^3 + 288*x^4 + 2192*x^5 +...
%e A240165 where
%e A240165 A(x) = 1/(1-x) + x/(1-3*x)^2 + x^2/(1-5*x)^3 + x^3/(1-7*x)^4 + x^4/(1-9*x)^5 +...
%t A240165 Table[Sum[Binomial[n,k] *(2*k+1)^(n-k),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 06 2014 *)
%t A240165 With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[2x])],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jul 17 2016 *)
%o A240165 (PARI) {a(n)=local(A=1);A=exp( x*(1 + exp(2*x +x*O(x^n))) );n!*polcoeff(A, n)}
%o A240165 for(n=0,30,print1(a(n),", "))
%o A240165 (PARI) {a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (2*k+1)*x +x*O(x^n))^(k+1));polcoeff(A, n)}
%o A240165 for(n=0,30,print1(a(n),", "))
%o A240165 (PARI) {a(n)=sum(k=0,n, binomial(n,k) * (2*k+1)^(n-k) )}
%o A240165 for(n=0,30,print1(a(n),", "))
%Y A240165 Cf. A080108, A216689.
%K A240165 nonn
%O A240165 0,2
%A A240165 _Paul D. Hanna_, Aug 02 2014