A266483 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).

1, 1, 9, 205, 8033, 456561, 34307545, 3219222301, 363018204225, 47866764942721, 7230829461286121, 1231746006983485005, 233652055492688836129, 48852757000944980067505, 11163401061821489604439737, 2768164393136241898192002781, 740339555234437428570144337025, 212438189627800855103688740374401, 65104233055709355841104275116309705, 21223353839635626633833547837080498509, 7333306933167926737746819644785091452641

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..110

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)^(4*n) * (x/N^3)^n/n! ]^(1/N).

(2) A(x) = exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(3*n+1)/n!] log( Sum_{n>=0} (n + y)^(4*n) * x^n/n! ). - Paul D. Hanna, Jan 29 2023

a(n) ~ 2^(3*n + 1/2) * (1 + sqrt(3))^(2*n-1) * exp((3-2*sqrt(3))*n - 4*sqrt(3) + 7) * n^(n-2) / 3^(3*n/2 + 1). - Vaclav Kotesovec, Mar 20 2024

Example

E.g.f.: A(x) = 1 + x + 9*x^2/2! + 205*x^3/3! + 8033*x^4/4! + 456561*x^5/5! + 34307545*x^6/6! + 3219222301*x^7/7! + 363018204225*x^8/8! + 47866764942721*x^9/9! + 7230829461286121*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^4*(x/N^3) + (N+2)^8*(x/N^3)^2/2! + (N+3)^12*(x/N^3)^3/3! + (N+4)^16*(x/N^3)^4/4! + (N+5)^20*(x/N^3)^5/5! + (N+6)^24*(x/N^3)^6/6! +...]^(1/N).

Prog

(PARI) /* Informal listing of terms 0..30 */
\p400
P(n) = sum(k=0, 32, (n+k)^(4*k) * x^k/k! +O(x^32))
Vec(round(serlaplace( subst(P(10^100)^(1/10^100), x, x/10^300) )*1.) )
(PARI) /* Using logarithmic formual */
{L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(4*m) *x^m/m! ) +x*O(x^n) ), n, x), 3*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

Cf. A266481, A266482, A266484, A266485, A266486.

Sequence in context: A103914 A211039 A307735 * A203364 A218887 A001535

Adjacent sequences: A266480 A266481 A266482 * A266484 A266485 A266486

Keyword

nonn

Author

Paul D. Hanna, Dec 30 2015

Status

approved