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A266481 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N).

%I A266481 #35 Mar 17 2024 11:12:06
%S A266481 1,1,5,55,993,25501,857773,35850795,1795564865,104972371417,
%T A266481 7022842421301,529428563641759,44421725002096225,4106744812439019765,
%U A266481 414834196219620026333,45462732300569936279251,5373006006732947705188737,681229881246574750274962225,92237589983019368975021777125,13283769418970268811752725081607,2027649185923009220298941142143201,326999803592314489529958494308640461,55558592280735155060861740192416874125
%N A266481 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N).
%C A266481 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 A266481 Conjecture: a(p*n) = 1 (mod p) for n>=0 and all prime p.
%H A266481 Paul D. Hanna, <a href="/A266481/b266481.txt">Table of n, a(n) for n = 0..100</a>
%F A266481 E.g.f. exp( Sum_{n>=0} A266526(n)*x^n/n! ), where A266526(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ).
%F A266481 a(n) ~ c * d^n * n^(n-2), where d = 2*(1 + sqrt(2)) * exp(1 - sqrt(2)) = 3.19091339076710837219515616759285808414857..., c = sqrt(1 - 1/sqrt(2)) * exp(3 - 2*sqrt(2)) = 0.642492128663019850313957348436... . - _Vaclav Kotesovec_, Jan 01 2016, updated Mar 17 2024
%e A266481 E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 25501*x^5/5! + 857773*x^6/6! + 35850795*x^7/7! + 1795564865*x^8/8! + 104972371417*x^9/9! + 7022842421301*x^10/10! +...
%e A266481 where A(x) equals the limit, as N -> oo, of the series
%e A266481 [1 + (N+1)^2*(x/N) + (N+2)^4*(x/N)^2/2! + (N+3)^6*(x/N)^3/3! + (N+4)^8*(x/N)^4/4! + (N+5)^10*(x/N)^5/5! + (N+6)^12*(x/N)^6/6! +...]^(1/N).
%e A266481 RELATED SERIES.
%e A266481 The following limit exists:
%e A266481 G(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ] / A(x)^N
%e A266481 where
%e A266481 G(x) = 1 + 2*x + 22*x^2/2! + 432*x^3/3! + 12220*x^4/4! + 451480*x^5/5! + 20591784*x^6/6! + 1117635008*x^7/7! + 70348179472*x^8/8! + 5037843612960*x^9/9! + 404453425948000*x^10/10! +...+ A266522(n)*x^n/n! +...
%e A266481 Logarithm of the g.f. A(x) begins:
%e A266481 Log(A(x)) = x + 4*x^2/2! + 42*x^3/3! + 752*x^4/4! + 19360*x^5/5! + 654912*x^6/6! + 27546736*x^7/7! + 1388207872*x^8/8! + 81621893376*x^9/9! + 5488951731200*x^10/10! +...+ A266526(n)*x^n/n! +...
%e A266481 and forms a diagonal in the triangles A266521 and A266488.
%o A266481 (PARI) {A266526(n) = n! * polcoeff( polcoeff( log( sum(m=0,n+1, (m + y)^(2*m) *x^m/m! ) +x*O(x^n) ),n,x), n+1,y)}
%o A266481 {a(n) = n! * polcoeff( exp( sum(m=1,n+1, A266526(m)*x^m/m! ) +x*O(x^n)), n)}
%o A266481 for(n=0,30,print1(a(n),", "))
%o A266481 (PARI) /* Informal listing of terms 0..30 */
%o A266481 \p100
%o A266481 P(n) = sum(k=0,31, (n+k)^(2*k) * x^k/k! +O(x^31))
%o A266481 Vec(round( serlaplace( subst(P(10^100)^(1/10^100),x,x/10^100) )*1.) )
%Y A266481 Cf. A266482, A266483, A266484, A266485, A266486, A266487, A266488, A266522, A266526.
%K A266481 nonn
%O A266481 0,3
%A A266481 _Paul D. Hanna_, Dec 30 2015