OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function
FORMULA
O.g.f.: Sum_{n>=0} x^n / (1 - (3*n+1)*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (3*k+1)^(n-k) for n>=0.
From Vaclav Kotesovec, Aug 06 2014: (Start)
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.
(a(n)/n!)^(1/n) ~ 3*exp(1/(2*LambertW(sqrt(3*n)/2))) / (2*LambertW(sqrt(3*n)/2)).
(End)
EXAMPLE
E.g.f.: E(x) = 1 + 2*x + 10*x^2/2! + 71*x^3/3! + 592*x^4/4! + 5777*x^5/5! +...
where E(x) = exp(x) * exp(x*exp(3*x)).
O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 71*x^3 + 592*x^4 + 5777*x^5 + 64792*x^6 +...
where
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 +...
MATHEMATICA
Table[Sum[Binomial[n, k] *(3*k+1)^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2014 *)
With[{nn=30}, CoefficientList[Series[Exp[x(1+Exp[3x])], {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Jun 09 2019 *)
PROG
(PARI) {a(n)=local(A=1); A=exp( x*(1 + exp(3*x +x*O(x^n))) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, (3*k+1)^(n-k)*binomial(n, k))}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 02 2014
STATUS
approved