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A266481 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N). 12
1, 1, 5, 55, 993, 25501, 857773, 35850795, 1795564865, 104972371417, 7022842421301, 529428563641759, 44421725002096225, 4106744812439019765, 414834196219620026333, 45462732300569936279251, 5373006006732947705188737, 681229881246574750274962225, 92237589983019368975021777125, 13283769418970268811752725081607, 2027649185923009220298941142143201, 326999803592314489529958494308640461, 55558592280735155060861740192416874125 (list; graph; refs; listen; history; text; internal format)
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!.

Conjecture: a(p*n) = 1 (mod p) for n>=0 and all prime p.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..100

FORMULA

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! ).

a(n) ~ c * d^n * n^(n-2), where d = 3.19091339076710837219515616759285808414857..., c = 0.642492128663019850313957348436... . - Vaclav Kotesovec, Jan 01 2016

EXAMPLE

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! +...

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

[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).

RELATED SERIES.

The following limit exists:

G(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ] / A(x)^N

where

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! +...

Logarithm of the g.f. A(x) begins:

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! +...

and forms a diagonal in the triangles A266521 and A266488.

PROG

(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)}

{a(n) = n! * polcoeff( exp( sum(m=1, n+1, A266526(m)*x^m/m! ) +x*O(x^n)), n)}

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

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

\p100

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

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

CROSSREFS

Cf. A266482, A266483, A266484, A266485, A266486, A266487, A266488, A266522, A266526.

Sequence in context: A135861 A141361 A203013 * A006150 A140049 A300589

Adjacent sequences:  A266478 A266479 A266480 * A266482 A266483 A266484

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 30 2015

STATUS

approved

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Last modified October 7 05:10 EDT 2022. Contains 357270 sequences. (Running on oeis4.)