A266484 -id:A266484 - cp's OEIS frontend

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

1, 1, 5, 55, 993, 25501, 857773, 35850795, 1795564865, 104972371417, 7022842421301, 529428563641759, 44421725002096225, 4106744812439019765, 414834196219620026333, 45462732300569936279251, 5373006006732947705188737, 681229881246574750274962225, 92237589983019368975021777125, 13283769418970268811752725081607, 2027649185923009220298941142143201, 326999803592314489529958494308640461, 55558592280735155060861740192416874125

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

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.

			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.

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

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

{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),", "))

/* 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.) )

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 = 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

A266483 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4n) (x/N^3)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 9, 205, 8033, 456561, 34307545, 3219222301, 363018204225, 47866764942721, 7230829461286121, 1231746006983485005, 233652055492688836129, 48852757000944980067505, 11163401061821489604439737, 2768164393136241898192002781, 740339555234437428570144337025, 212438189627800855103688740374401, 65104233055709355841104275116309705, 21223353839635626633833547837080498509, 7333306933167926737746819644785091452641

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

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!.

			E.g.f.: A(x) = 1 + x + 9*x^2/2! + 205*x^3/3! + 8033*x^4/4! + 456561*x^5/5! + 34307545*x^6/6! + 3219222301*x^7/7! + 363018204225*x^8/8! + 47866764942721*x^9/9! + 7230829461286121*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^4*(x/N^3) + (N+2)^8*(x/N^3)^2/2! + (N+3)^12*(x/N^3)^3/3! + (N+4)^16*(x/N^3)^4/4! + (N+5)^20*(x/N^3)^5/5! + (N+6)^24*(x/N^3)^6/6! +...]^(1/N).

Vaclav Kotesovec, Table of n, a(n) for n = 0..110

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

/* Informal listing of terms 0..30 */
\p400
P(n) = sum(k=0,32, (n+k)^(4*k) * x^k/k! +O(x^32))
Vec(round(serlaplace( subst(P(10^100)^(1/10^100),x,x/10^300) )*1.) )

/* Using logarithmic formual */
{L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(4*m) *x^m/m! ) +x*O(x^n) ), n, x), 3*n+1, y)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 29 2023

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(3*n+1)/n!] log( Sum_{n>=0} (n + y)^(4*n) * x^n/n! ). - Paul D. Hanna, Jan 29 2023
a(n) ~ 2^(3*n + 1/2) * (1 + sqrt(3))^(2*n-1) * exp((3-2*sqrt(3))*n - 4*sqrt(3) + 7) * n^(n-2) / 3^(3*n/2 + 1). - Vaclav Kotesovec, Mar 20 2024

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

Original entry on oeis.org

1, 1, 7, 118, 3373, 139096, 7565779, 513277024, 41820455065, 3982842285184, 434457816912991, 53434112376345856, 7317518431787267653, 1104465712210096168960, 182183636400541105459627, 32609250878782525222260736, 6295153043394143761311198769, 1303848990485145459272159297536, 288415207140946760926622987982775, 67863051757810284274576363569872896, 16924929956887283486906002826128780381, 4459845456377312896416211474995205636096

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

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!.

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..120

Cf. A266481, A266483, A266484, A266485, A266486, A266523, A266524, A266525.

/* Informal listing of terms 0..30 */
\p200
P(n) = sum(k=0, 31, (n+k)^(3*k) * x^k/k! +O(x^31))
Vec(round( serlaplace( subst(P(10^100)^(1/10^100), x, x/10^200) )*1.) )

{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)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 15 2021

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

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

A266485 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 2n)^(2n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 9, 193, 6929, 356001, 24004825, 2012327521, 202156421409, 23701550853313, 3179302948594601, 480443117415138945, 80788534008942735409, 14965275494082095616097, 3028424508967743713615481, 664790043100841638943719201, 157352199248412053285546165825, 39950540529265571984889165180801

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

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!.
Related limits (Paul D. Hanna, Jan 20 2023):
exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
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).

			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! +...
where A(x) equals the limit, as N -> oo, of the series
[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).
The logarithm of the g.f. A(x) begins (_Paul D. Hanna_, Jan 20 2023):
(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! + ...
where A359926(n) = [x^n*y^(n+1)/n!] (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! );
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
(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! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..200

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

/* Informal listing of terms 0..30 */
\p300
P(n) = sum(k=0,32, (n+2*k)^(2*k) * x^k/k! +O(x^31))
Vec( round( serlaplace( subst(P(10^100)^(1/10^100),x,x/10^100) )*1.) )

/* Using formula for the logarithm of g.f. A(x) Paul D. Hanna, Jan 20 2023 */
{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)}
{a(n) = n! * polcoeff( exp( sum(m=1, n+1, L(m)*x^m/m! ) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

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

A266486 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 3n)^(2n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 13, 415, 22321, 1721101, 174252997, 21935478979, 3308902366945, 582483654850105, 117302814498577501, 26610247617703733479, 6716634535536518884177, 1867456548257171896034245, 567177496490226897535216405, 186852683125922747089699211851, 66371163246016212237620717414593, 25287323016605747194753141853886961, 10287301449354981886046538248627595565, 4450859089975905722184296672608494825775, 2040775907870521098252331495354110194770801

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

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!.

			E.g.f.: A(x) = 1 + x + 13*x^2/2! + 415*x^3/3! + 22321*x^4/4! + 1721101*x^5/5! + 174252997*x^6/6! + 21935478979*x^7/7! + 3308902366945*x^8/8! + 582483654850105*x^9/9! + 117302814498577501*x^10/10! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+3)^2*(x/N) + (N+6)^4*(x/N)^2/2! + (N+9)^6*(x/N)^3/3! + (N+12)^8*(x/N)^4/4! + (N+15)^10*(x/N)^5/5! + (N+18)^12*(x/N)^6/6! +...]^(1/N).

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

/* Informal listing of terms 0..30 */
\p300
P(n) = sum(k=0,32, (n+3*k)^(2*k) * x^k/k! +O(x^32))
Vec( round( serlaplace( subst(P(10^100)^(1/10^100),x,x/10^100) )*1.) )

A266487 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N - n)^(2n) (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, -3, 31, -559, 14541, -496811, 21081859, -1070585055, 63366015673, -4285932328819, 326248732427751, -27610580638457807, 2572239828612623365, -261621661000490429211, 28849626308051995285771, -3428690784657696770593471, 436924188109882619766249201, -59432725217403244945921112675, 8595527924368773785463788378287, -1317123285394547040368548520041839, 213171869078193696050387803319525821, -36338236299957647745418230431675850763, 6507698606647750492700809995200106342675, -1221579456277487714539848255959245396739999

Offset: 0

Paul D. Hanna, Dec 30 2015

sign

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!.

			E.g.f.: A(x) = 1 + x - 3*x^2/2! + 31*x^3/3! - 559*x^4/4! + 14541*x^5/5! - 496811*x^6/6! + 21081859*x^7/7! - 1070585055*x^8/8! + 63366015673*x^9/9! - 4285932328819*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).

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

/* Informal listing of terms 0..30 */
\p300
H(n) = sum(k=0,32, (n - k)^(2*k) * x^k/k! +O(x^32))
Vec( round( serlaplace( subst(H(10^100)^(1/10^100),x,x/10^100) )*1.) )

A266525 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5n) (x/N^4)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5n) (x/N^4)^n/n! ]^(1/N).

Original entry on oeis.org

1, 5, 160, 9135, 750400, 80441425, 10638828000, 1673678753075, 305252823558400, 63325918470124125, 14724939203560768000, 3793154255510116564375, 1072236911373050595840000, 329985748809343574149723625, 109830285822698899619230720000, 39309730439858456963398059166875, 15055402080033663459327206195200000, 6143747797144623366547686616298003125, 2661215654340427415860408455902822400000, 1219479030123689259752174147774198563109375, 589404548968234611551047396687998740070400000, 299658512455145134987556717044427762586006890625, 159865819819818837465659104892463315321094144000000

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

The e.g.f. A(x) of this sequence also satisfies:
A(x*y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(5*n) * (x/N^4)^n/n! ] / G(x,y)^N
where
G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(5*n) * (x/N^4)^n/n! ]^(1/N)
for all real y.

			E.g.f.: A(x) = 1 + 5*x + 160*x^2/2! + 9135*x^3/3! + 750400*x^4/4! + 80441425*x^5/5! + 10638828000*x^6/6! + 1673678753075*x^7/7! + 305252823558400*x^8/8! + 63325918470124125*x^9/9! + 14724939203560768000*x^10/10! +...
such that
A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ] / F(x)^N
where
F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N)
and
F(x) = 1 + x + 11*x^2/2! + 316*x^3/3! + 15741*x^4/4! + 1140376*x^5/5! + 109350271*x^6/6! + 13100626176*x^7/7! + 1886686497401*x^8/8! + 317762099341696*x^9/9! + 61318533545522451*x^10/10! +...+ A266484(n)*x^n/n! +...

Cf. A266484, A266522, A266523, A266524.

A360341 a(n) = coefficient of x^ny^(3n+1)/n! in log( Sum_{n>=0} (n + y)^(5n) x^n/n! ).

Original entry on oeis.org

1, 10, 285, 14240, 1036225, 99774720, 11995938325, 1732780710400, 292580972777025, 56581144474976000, 12335796889894262125, 2994228576573719040000, 800930404887937807458625, 234113078032084301026816000, 74248479783538967821383793125, 25394786139647229685682094080000

Offset: 1

Paul D. Hanna, Feb 10 2023

nonn

			E.g.f.: A(x) = x + 10*x^2/2! + 285*x^3/3! + 14240*x^4/4! + 1036225*x^5/5! + 99774720*x^6/6! + 11995938325*x^7/7! + 1732780710400*x^8/8! + ... + a(n)*x^n/n! + ...
where a(n) equals the coefficient of y^(4*n+1)*x^n/n! in the series given by
log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ) = (y^5 + 5*y^4 + 10*y^3 + 10*y^2 + 5*y + 1)*x + (10*y^9 + 135*y^8 + 840*y^7 + 3150*y^6 + 7812*y^5 + 13230*y^4 + 15240*y^3 + 11475*y^2 + 5110*y + 1023)*x^2/2! + (285*y^13 + 6985*y^12 + 82800*y^11 + 626640*y^10 + 3365015*y^9 + 13480875*y^8 + 41269545*y^7 + 97340225*y^6 + 176218089*y^5 + 241023105*y^4 + 241403365*y^3 + 167262045*y^2 + 71713845*y + 14345837)*x^3/3! + (14240*y^17 + 535150*y^16 + 9965360*y^15 + 121806600*y^14 + 1090732800*y^13 + 7563031080*y^12 + 41870604200*y^11 + 188252006020*y^10 + 693127766960*y^9 + 2094270509580*y^8 + 5176075514880*y^7 + 10375810342800*y^6 + 16622405553984*y^5 + 20792525880990*y^4 + 19576849364160*y^3 + 13053873999580*y^2 + 5496952909520*y + 1099451098702)*x^4/4! + ...
Exponentiation yields the e.g.f. of A266484:
exp(A(x)) = 1 + x + 11*x^2/2! + 316*x^3/3! + 15741*x^4/4! + 1140376*x^5/5! + 109350271*x^6/6! + 13100626176*x^7/7! + 1886686497401*x^8/8! + ... + A266484(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 5, 95, 3560, 207245, 16629120, 1713705475, 216597588800, ...].

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A266484, A360339, A360340.

/* Using logarithmic formula */
{a(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m + y)^(5*m) *x^m/m! ) +x*O(x^n) ), n, x), 4*n+1, y)}
for(n=1, 20, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! may be defined as follows.
(1) A(x) = Limit_{N->oo} (1/N) * log( Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ).
(2) a(n) = [x^n*y^(3*n+1)/n!] log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = (25/16) * (5 + 2*sqrt(5)) * exp(5 - 2*sqrt(5)) = 25.090908742294025045771061662375185533388200826641029119554... and c = 1/(8*sqrt((1 + 2/sqrt(5))*Pi)) = 0.05123846578813482717849518499100286... - Vaclav Kotesovec, Feb 12 2023, updated Mar 20 2024

A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + ny)^(2n) * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Dec 30 2015

nonn

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!.

			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; ...

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

/* 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))))

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

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A266483 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).

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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).

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A266485 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).

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A266486 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 3*n)^(2*n) * (x/N)^n/n! ]^(1/N).

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A266487 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N - n)^(2*n) * (x/N)^n/n! ]^(1/N).

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A266525 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).

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A360341 a(n) = coefficient of x^n*y^(3*n+1)/n! in log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ).

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

A266483 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4n) (x/N^3)^n/n! ]^(1/N).

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

A266485 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 2n)^(2n) * (x/N)^n/n! ]^(1/N).

A266486 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + 3n)^(2n) * (x/N)^n/n! ]^(1/N).

A266487 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N - n)^(2n) (x/N)^n/n! ]^(1/N).

A266525 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5n) (x/N^4)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5n) (x/N^4)^n/n! ]^(1/N).

A360341 a(n) = coefficient of x^ny^(3n+1)/n! in log( Sum_{n>=0} (n + y)^(5n) x^n/n! ).

A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + ny)^(2n) * (x/N)^n/n! ]^(1/N).