OFFSET
0,3
COMMENTS
Compare to: Limit_{N->oo} [ Sum_{n>=0} (N + n)^n * x^n/n! ]^(1/N) = Sum_{n>=0} (n+1)^(n-1) * x^n/n!.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..100
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(4*n+1)/n!] log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ). - Paul D. Hanna, Jan 29 2023
a(n) ~ 5^(5*n/2 + 1/4) * (1 + sqrt(5))^(3*n - 3/2) * exp((4-2*sqrt(5))*n - 4*sqrt(5) + 9) * n^(n-2) / 2^(7*n + 1). - Vaclav Kotesovec, Mar 20 2024
EXAMPLE
E.g.f.: A(x) = 1 + x + 11*x^2/2! + 316*x^3/3! + 15741*x^4/4! + 1140376*x^5/5! + 109350271*x^6/6! + 13100626176*x^7/7! + 1886686497401*x^8/8! + 317762099341696*x^9/9! + 61318533545522451*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^5*(x/N^4) + (N+2)^10*(x/N^4)^2/2! + (N+3)^15*(x/N^4)^3/3! + (N+4)^20*(x/N^4)^4/4! + (N+5)^25*(x/N^4)^5/5! + (N+6)^30*(x/N^4)^6/6! +...]^(1/N).
PROG
(PARI) /* Informal listing of terms 0..30 */
\p500
P(n) = sum(k=0, 32, (n+k)^(5*k) * x^k/k! +O(x^32))
Vec(round(serlaplace( subst(P(10^100)^(1/10^100), x, x/10^400) )*1.) )
(PARI) /* Using logarithmic formula */
{L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(5*m) *x^m/m! ) +x*O(x^n) ), n, x), 4*n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 29 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 30 2015
STATUS
approved