A266485 - OEIS

%I #24 Mar 17 2024 09:50:42

%S 1,1,9,193,6929,356001,24004825,2012327521,202156421409,

%T 23701550853313,3179302948594601,480443117415138945,

%U 80788534008942735409,14965275494082095616097,3028424508967743713615481,664790043100841638943719201,157352199248412053285546165825,39950540529265571984889165180801

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

%C 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!.

%C Related limits (_Paul D. Hanna_, Jan 20 2023):

%C exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).

%C W(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n)^n * (x/N)^n/n! ]^(1/N), where W(x) = LambertW(-x)/(-x).

%H Paul D. Hanna, <a href="/A266485/b266485.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following (_Paul D. Hanna_, Jan 20 2023):

%F (1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).

%F (2) A(x) = exp( Sum_{n>=0} A359926(n)*x^n/n! ), where A359926(n) = (1/4) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + 2*y)^(2*n) *x^n/n! ).

%F a(n) ~ c * d^n * n^(n-2), where d = 4*(1 + sqrt(2)) * exp(1 - sqrt(2)) = 6.3818267815342167443903123351857161682971406064645602440616... and c = sqrt(1 - 1/sqrt(2))/2 * exp(3/2 - sqrt(2)) = 0.294836494691148677397464568534316405253091784834436235... - _Vaclav Kotesovec_, Jan 21 2023, updated Mar 17 2024

%e E.g.f.: A(x) = 1 + x + 9*x^2/2! + 193*x^3/3! + 6929*x^4/4! + 356001*x^5/5! + 24004825*x^6/6! + 2012327521*x^7/7! + 202156421409*x^8/8! + 23701550853313*x^9/9! + 3179302948594601*x^10/10! +...

%e where A(x) equals the limit, as N -> oo, of the series

%e [1 + (N+2)^2*(x/N) + (N+4)^4*(x/N)^2/2! + (N+6)^6*(x/N)^3/3! + (N+8)^8*(x/N)^4/4! + (N+10)^10*(x/N)^5/5! + (N+12)^12*(x/N)^6/6! +...]^(1/N).

%e The logarithm of the g.f. A(x) begins (_Paul D. Hanna_, Jan 20 2023):

%e (a) log(A(x)) = x + 8*x^2/2! + 168*x^3/3! + 6016*x^4/4! + 309760*x^5/5! + 20957184*x^6/6! + 1762991104*x^7/7! + 177690607616*x^8/8! + ... + A359926(n)*x^n/n! + ...

%e where A359926(n) = [x^n*y^(n+1)/n!] (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! );

%e that is, the coefficient of x^n/n! in the logarithm of e.g.f A(x) equals the coefficient of y^(n+1)*x^n/n! in the series given by

%e (b) (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ) = (y^2 + y + 1/4)*x + (8*y^3 + 18*y^2 + 14*y + 15/4)*x^2/2! + (168*y^4 + 632*y^3 + 933*y^2 + 639*y + 683/4)*x^3/3! + (6016*y^5 + 33088*y^4 + 76480*y^3 + 92680*y^2 + 58720*y + 31019/2)*x^4/4! + ...

%o (PARI) /* Informal listing of terms 0..30 */

%o \p300

%o P(n) = sum(k=0,32, (n+2*k)^(2*k) * x^k/k! +O(x^31))

%o Vec( round( serlaplace( subst(P(10^100)^(1/10^100),x,x/10^100) )*1.) )

%o (PARI) /* Using formula for the logarithm of g.f. A(x) _Paul D. Hanna_, Jan 20 2023 */

%o {L(n) = (1/4) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + 2*y)^(2*m) *x^m/m! ) +x*O(x^n) ), n, x), n+1, y)}

%o {a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A266481, A266482, A266483, A266484, A266486, A266487, A359926, A359927, A319147, A318633, A319834.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 30 2015