A245834 - OEIS

%I #14 Jun 09 2019 15:56:51

%S 1,2,10,71,592,5777,64792,814025,11264176,169871633,2768582104,

%T 48412950929,902831609368,17865749820089,373564063839376,

%U 8223263706957713,189960800250512608,4591950749700004385,115866075506169417256,3044877330738661504625,83169542349597382767496,2356949307613191494567561

%N E.g.f.: exp( x*(1 + exp(3*x)) ).

%H Vincenzo Librandi, <a href="/A245834/b245834.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 - (3*n+1)*x)^(n+1).

%F a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+1)^(n-k) for n>=0.

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

%F a(n) ~ exp((1+exp(3*r))*r - n) * n^(n+1/2) / (r^n * sqrt(r + exp(3*r)*r* (1+9*r*(1+r)))), where r is the root of the equation r*(1 + exp(3*r) + 3*r*exp(3*r)) = n.

%F (a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)).

%F (End)

%e E.g.f.: E(x) = 1 + 2*x + 10*x^2/2! + 71*x^3/3! + 592*x^4/4! + 5777*x^5/5! +...

%e where E(x) = exp(x) * exp(x*exp(3*x)).

%e O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 71*x^3 + 592*x^4 + 5777*x^5 + 64792*x^6 +...

%e where

%e A(x) = 1/(1-x) + x/(1-4*x)^2 + x^2/(1-7*x)^3 + x^3/(1-10*x)^4 + x^4/(1-13*x)^5 +...

%t Table[Sum[Binomial[n,k] *(3*k+1)^(n-k),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Aug 06 2014 *)

%t With[{nn=30},CoefficientList[Series[Exp[x(1+Exp[3x])],{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jun 09 2019 *)

%o (PARI) {a(n)=local(A=1);A=exp( x*(1 + exp(3*x +x*O(x^n))) );n!*polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - (3*k+1)*x +x*O(x^n))^(k+1));polcoeff(A, n)}

%o for(n=0,30,print1(a(n),", "))

%o (PARI) {a(n)=sum(k=0,n,(3*k+1)^(n-k)*binomial(n,k))}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A080108, A216689, A240165, A245835.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 02 2014