This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A266483 #15 Mar 20 2024 09:20:16
%S A266483 1,1,9,205,8033,456561,34307545,3219222301,363018204225,
%T A266483 47866764942721,7230829461286121,1231746006983485005,
%U A266483 233652055492688836129,48852757000944980067505,11163401061821489604439737,2768164393136241898192002781,740339555234437428570144337025,212438189627800855103688740374401,65104233055709355841104275116309705,21223353839635626633833547837080498509,7333306933167926737746819644785091452641
%N A266483 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).
%C A266483 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!.
%H A266483 Vaclav Kotesovec, <a href="/A266483/b266483.txt">Table of n, a(n) for n = 0..110</a>
%F A266483 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
%F A266483 (1) A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).
%F A266483 (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
%F A266483 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
%e A266483 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! +...
%e A266483 where A(x) equals the limit, as N -> oo, of the series
%e A266483 [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).
%o A266483 (PARI) /* Informal listing of terms 0..30 */
%o A266483 \p400
%o A266483 P(n) = sum(k=0,32, (n+k)^(4*k) * x^k/k! +O(x^32))
%o A266483 Vec(round(serlaplace( subst(P(10^100)^(1/10^100),x,x/10^300) )*1.) )
%o A266483 (PARI) /* Using logarithmic formual */
%o A266483 {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)}
%o A266483 {a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
%o A266483 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jan 29 2023
%Y A266483 Cf. A266481, A266482, A266484, A266485, A266486.
%K A266483 nonn
%O A266483 0,3
%A A266483 _Paul D. Hanna_, Dec 30 2015