login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N). 1
1, 1, 0, 1, 4, 0, 1, 12, 42, 0, 1, 24, 216, 752, 0, 1, 40, 660, 5440, 19360, 0, 1, 60, 1560, 22320, 178920, 654912, 0, 1, 84, 3150, 68320, 916440, 7316064, 27546736, 0, 1, 112, 5712, 173600, 3432800, 44693376, 359051392, 1388207872, 0, 1, 144, 9576, 387072, 10493280, 197261568, 2536797312, 20605529088, 81621893376, 0, 1, 180, 15120, 782880, 27735120, 702777600, 12845683200, 164732083200, 1355581612800, 5488951731200, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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!.

LINKS

Table of n, a(n) for n=0..65.

EXAMPLE

E.g.f. A(x) = 1 + x +

x^2/2! * (1 + 4*y) +

x^3/3! * (1 + 12*y + 42*y^2) +

x^4/4! * (1 + 24*y + 216*y^2 + 752*y^3) +

x^5/5! * (1 + 40*y + 660*y^2 + 5440*y^3 + 19360*y^4) +

x^6/6! * (1 + 60*y + 1560*y^2 + 22320*y^3 + 178920*y^4 + 654912*y^5) +

x^7/7! * (1 + 84*y + 3150*y^2 + 68320*y^3 + 916440*y^4 + 7316064*y^5 + 27546736*y^6) +

x^8/8! * (1 + 112*y + 5712*y^2 + 173600*y^3 + 3432800*y^4 + 44693376*y^5 + 359051392*y^6 + 1388207872*y^7) + ...

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

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

Triangle of coefficients T(n,k) of x^n*y^k/n!, n>=0, k=0..n, begins:

1;

1, 0;

1, 4, 0;

1, 12, 42, 0;

1, 24, 216, 752, 0;

1, 40, 660, 5440, 19360, 0;

1, 60, 1560, 22320, 178920, 654912, 0;

1, 84, 3150, 68320, 916440, 7316064, 27546736, 0;

1, 112, 5712, 173600, 3432800, 44693376, 359051392, 1388207872, 0;

1, 144, 9576, 387072, 10493280, 197261568, 2536797312, 20605529088, 81621893376, 0;

1, 180, 15120, 782880, 27735120, 702777600, 12845683200, 164732083200, 1355581612800, 5488951731200, 0;

1, 220, 22770, 1467840, 65659440, 2143842624, 52117998240, 938463651840, 12065358919680, 100649306644480, 415721105434624, 0; ...

PROG

(PARI) /* Print the initial rows of this triangle: */

\p400

P(n) = sum(k=0, 21, (n + k*y)^(2*k) * (x/n)^k/k! +O(x^21))

V=Vec( round( serlaplace( P(10^100)^(1/10^100) )*1.) )

for(n=1, 15, print(Vec(V[n]+O(y^n))))

CROSSREFS

Cf. A266481, A266482, A266483, A266484, A266485, A266486, A266487.

Sequence in context: A178979 A228270 A335748 * A189355 A221817 A054375

Adjacent sequences:  A266485 A266486 A266487 * A266489 A266490 A266491

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 30 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 17 20:44 EDT 2021. Contains 343990 sequences. (Running on oeis4.)