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A266482 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N).

%I A266482 #22 Mar 20 2024 09:19:40
%S A266482 1,1,7,118,3373,139096,7565779,513277024,41820455065,3982842285184,
%T A266482 434457816912991,53434112376345856,7317518431787267653,
%U A266482 1104465712210096168960,182183636400541105459627,32609250878782525222260736,6295153043394143761311198769,1303848990485145459272159297536,288415207140946760926622987982775,67863051757810284274576363569872896,16924929956887283486906002826128780381,4459845456377312896416211474995205636096
%N A266482 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N).
%C A266482 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 A266482 Vaclav Kotesovec, <a href="/A266482/b266482.txt">Table of n, a(n) for n = 0..120</a>
%F A266482 E.g.f. exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(2*n+1)/n!] log( Sum_{n>=0} (n + y)^(3*n) * x^n/n! ). - _Paul D. Hanna_, Jul 15 2021
%F A266482 a(n) ~ 3^(n + 1/2) * (3 + sqrt(6))^(n - 1/2) * exp((2-sqrt(6))*n - 2*sqrt(6) + 5) * n^(n-2) / 2^(n + 3/2). - _Vaclav Kotesovec_, Mar 20 2024
%e A266482 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 118*x^3/3! + 3373*x^4/4! + 139096*x^5/5! + 7565779*x^6/6! + 513277024*x^7/7! + 41820455065*x^8/8! + 3982842285184*x^9/9! + 434457816912991*x^10/10! +...
%e A266482 where A(x) equals the limit, as N -> oo, of the series
%e A266482 [1 + (N+1)^3*(x/N^2) + (N+2)^6*(x/N^2)^2/2! + (N+3)^9*(x/N^2)^3/3! + (N+4)^12*(x/N^2)^4/4! + (N+5)^15*(x/N^2)^5/5! + (N+6)^18*(x/N^2)^6/6! +...]^(1/N).
%o A266482 (PARI) /* Informal listing of terms 0..30 */
%o A266482 \p200
%o A266482 P(n) = sum(k=0, 31, (n+k)^(3*k) * x^k/k! +O(x^31))
%o A266482 Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^200) )*1.) )
%o A266482 (PARI) {L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(3*m) *x^m/m! ) +x*O(x^n) ), n, x), 2*n+1, y)}
%o A266482 {a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
%o A266482 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jul 15 2021
%Y A266482 Cf. A266481, A266483, A266484, A266485, A266486, A266523, A266524, A266525.
%K A266482 nonn
%O A266482 0,3
%A A266482 _Paul D. Hanna_, Dec 30 2015