A266481 -id:A266481 - cp's OEIS frontend

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

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

A266484 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5n) (x/N^4)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 11, 316, 15741, 1140376, 109350271, 13100626176, 1886686497401, 317762099341696, 61318533545522451, 13343942849386863616, 3233753469962945660341, 863794149132594286734336, 252178372791563562485494151, 79890921514691257167186558976, 27298165065421976828646695794161, 10007689235634878438090676073824256, 3918413783588692571816707646546345371, 1631982989611299844119224469019967225856, 720447625733586591482575137323090206302701

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 + 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! +...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N+1)^5*(x/N^4) + (N+2)^10*(x/N^4)^2/2! + (N+3)^15*(x/N^4)^3/3! + (N+4)^20*(x/N^4)^4/4! + (N+5)^25*(x/N^4)^5/5! + (N+6)^30*(x/N^4)^6/6! +...]^(1/N).

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

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

/* Informal listing of terms 0..30 */
\p500
P(n) = sum(k=0,32, (n+k)^(5*k) * x^k/k! +O(x^32))
Vec(round(serlaplace( subst(P(10^100)^(1/10^100),x,x/10^400) )*1.) )

/* Using logarithmic formula */
{L(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)}
{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)^(5*n) * (x/N^4)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} L(n)*x^n/n! ), where L(n) = [x^n*y^(4*n+1)/n!] log( Sum_{n>=0} (n + y)^(5*n) * x^n/n! ). - Paul D. Hanna, Jan 29 2023
a(n) ~ 5^(5*n/2 + 1/4) * (1 + sqrt(5))^(3*n - 3/2) * exp((4-2*sqrt(5))*n - 4*sqrt(5) + 9) * n^(n-2) / 2^(7*n + 1). - 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.) )

A266526 a(n) = coefficient of x^ny^(n+1)/n! in Log( Sum_{n>=0} (n + y)^(2n) * x^n/n! ), for n>=1.

Original entry on oeis.org

1, 4, 42, 752, 19360, 654912, 27546736, 1388207872, 81621893376, 5488951731200, 415721105434624, 35026876903256064, 3250356630453317632, 329437813126362185728, 36214170617862339840000, 4291812357982293898231808, 545518054282041342531076096, 74032137722410904128877494272, 10684317262536125210489796296704, 1634019721630446295055397683200000

Offset: 1

Paul D. Hanna, Jan 01 2016

nonn

Equals the logarithm of the e.g.f. of A266481.

Equals the right border of triangle A266521.

			E.g.f: 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! +...
where exponentiation yields the e.g.f. of A266481:
exp(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! +...+ A266481(n)*x^n/n! +...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N).

Vaclav Kotesovec, Table of n, a(n) for n = 1..230

Cf. A266521, A266481.

{a(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)}
for(n=1,30, print1(a(n),", "))

a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) * exp((1 - sqrt(2))*n) * n^(n-2). - Vaclav Kotesovec, Mar 20 2024

A318633 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 1, 7, 25, 541, 3361, 135451, 1179697, 72062425, 800549281, 66556219951, 895033831561, 94883671398517, 1494100965599425, 193406329375101571, 3484870135007879521, 534176530923146533681, 10828212525851495841217, 1921147013664776094354775, 43243444110218718242048761, 8725818848745836121239731021, 215849236633217233817102676961, 48851798675231712377778383815147

Offset: 0

Paul D. Hanna, Sep 04 2018

nonn

Compare to:
(C1) exp(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = Limit_{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 + x^2/2! + 7*x^3/3! + 25*x^4/4! + 541*x^5/5! + 3361*x^6/6! + 135451*x^7/7! + 1179697*x^8/8! + 72062425*x^9/9! + 800549281*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+1)*(x/N) + (N^2+2^2)^2*(x/N)^2/2! + (N^2+3^2)^3*(x/N)^3/3! + (N^2+4^2)^4*(x/N)^4/4! + (N^2+5^2)^5*(x/N)^5/5! + (N^2+6^2)^6*(x/N)^6/6! +...]^(1/N).
RELATED SERIES.
(a) The following limit exists:
G(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ] / A(x)^N
where G(x) is an even function that begins
G(x) = 1 + 6*x^2/2! + 636*x^4/4! + 200520*x^6/6! + 127426320*x^8/8! + 135927439200*x^10/10! + 218711489895360*x^12/12! + 494824278347118720*x^14/14! 1497827553156635500800*x^16/16! +...
(b) The logarithm of the e.g.f. A(x) is an odd function that begins:
Log(A(x)) = x + 6*x^3/3! + 480*x^5/5! + 122640*x^7/7! + 66044160*x^9/9! + 61482516480*x^11/11! + 88135315107840*x^13/13! + 180378921026304000*x^15/15! + ... + A318634(n)*x^(2*n-1)/(2*n-1)! +...

Paul D. Hanna, Table of n, a(n) for n = 0..150

Cf. A318634, A266481.

{A318634(n) = 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, A318634(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^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.) )

E.g.f. exp( Sum_{n>=0} A318634(n)*x^n/n! ), where A318634(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + y^2)^n *x^n/n! ).

a(n) ~ c * 2^(3*n/2) * (1 + sqrt(5))^(n/2) * exp((3 - sqrt(5))*n/2) * n^(n-2) / (2^(n-2) * exp(n)), where c = 0.06299092342072472381892574307... if n is even and c = 0.21603924353200238368086490887... if n is odd. - Vaclav Kotesovec, Mar 21 2024

A319147 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + Nn + n^2)^n (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 3, 22, 269, 4776, 111967, 3280264, 115550073, 4762181440, 224954474651, 11987717900544, 711604917300037, 46572971758429312, 3332107859592406455, 258748811312125854976, 21674785904235983431793, 1948303837796264786497536, 187062919027712092164076723, 19107058023481400501276569600

Offset: 0

Paul D. Hanna, Sep 19 2018

nonn

Compare to:
(C1) exp(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) W(x) = Limit_{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 + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! + 4762181440*x^9/9! + 224954474651*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+N+1)*(x/N) + (N^2+2*N+2^2)^2*(x/N)^2/2! + (N^2+3*N+3^2)^3*(x/N)^3/3! + (N^2+4*N+4^2)^4*(x/N)^4/4! + (N^2+5*N+5^2)^5*(x/N)^5/5! + (N^2+6*N+6^2)^6*(x/N)^6/6! +...]^(1/N).
RELATED SERIES.
(a) The logarithm of the g.f. A(x) begins:
log(A(x)) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ... + A319834(n)*x^n/n! + ...
where A319834(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! );
that is, the coefficients in the logarithm of e.g.f A(x) equals the coefficients of y^(n+1)*x^n/n! in the series given by
(b) log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ) = (y^2 + y + 1)*x + (2*y^3 + 9*y^2 + 14*y + 15)*x^2/2! + (15*y^4 + 107*y^3 + 366*y^2 + 639*y + 683)*x^3/3! + (184*y^5 + 2038*y^4 + 10432*y^3 + 32308*y^2 + 58720*y + 62038)*x^4/4! + (3325*y^6 + 50469*y^5 + 367155*y^4 + 1636590*y^3 + 4833195*y^2 + 8940045*y + 9342629)*x^5/5! + (79056*y^7 + 1565256*y^6 + 15015936*y^5 + 90978240*y^4 + 376955520*y^3 + 1085556216*y^2 + 2027376336*y + 2100483216)*x^6/6! + ...
(c) The following limit exists
G(x) = Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ] / A(x)^N
where
G(x) = 1 + x + 10*x^2/2! + 135*x^3/3! + 2764*x^4/4! + 72665*x^5/5! + 2362896*x^6/6! + 91282975*x^7/7! + 4088186320*x^8/8! + 208223576721*x^9/9! + ...
the logarithm of which begins
log(G(x)) = x + 9*x^2/2! + 107*x^3/3! + 2038*x^4/4! + 50469*x^5/5! + 1565256*x^6/6! + 58095463*x^7/7! + 2513768496*x^8/8! + ... + D(n)*x^n/n! + ...
where D(n) = [x^n*y^n/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n * x^n/n! ).

Vaclav Kotesovec, Table of n, a(n) for n = 0..150 (terms 0..100 from Paul D. Hanna)

Cf. A266481, A318633, A319834.

{L(n) = n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + 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), ", "))

/* Informal listing of terms 0..30 */
\p100
P(n) = sum(k=0, 31, (n^2 + n*k + 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} L(n)*x^n/n! ), where L(n) = [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + y^2)^n *x^n/n! ).

a(n) ~ c * d^n * n! / n^(5/2), where d = 6.16018341007619464488... and c = 0.240315927519139896... - Vaclav Kotesovec, Mar 19 2022

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.) )

A266522 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2n) (x/N)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2n) (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 2, 22, 432, 12220, 451480, 20591784, 1117635008, 70348179472, 5037843612960, 404453425948000, 35977638091065088, 3512312454013520832, 373346162796913784192, 42922941487808176036480, 5307003951337894697856000, 702183042248318469458657536, 98997224309112273722486891008, 14815674464782854979394204308992, 2345767767928443601985964232355840, 391750020994050554579656281189760000, 68820978855281989513379320801711429632

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)^(2*n) * (x/N)^n/n! ] / G(x,y)^N
where
G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (x/N)^n/n! ]^(1/N)
for all real y.

			E.g.f.: A(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! + ...
such that
A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ] / F(x)^N
where
F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2*n) * (x/N)^n/n! ]^(1/N)
and
F(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! +...+ A266481(n)*x^n/n! + ...
RELATED SERIES.
log(A(x)) = 2*x + 18*x^2/2! + 316*x^3/3! + 8272*x^4/4! + 288048*x^5/5! + 12523584*x^6/6! + 652959872*x^7/7! + 39701769216*x^8/8! + 2758053332736*x^9/9! + ... + A266521(n,n)*x^n/n! + ...

Cf. A266481, A266523, A266524, A266525, A266521.

E.g.f.: exp( Sum_{n>=1} A266521(n,n)*x^n/n! ), where the e.g.f. of triangle A266521 is Log(Sum_{n>=0} (n + y)^(2*n) * x^n/n!). - Paul D. Hanna, Sep 30 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A266484 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5*n) * (x/N^4)^n/n! ]^(1/N).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A266526 a(n) = coefficient of x^n*y^(n+1)/n! in Log( Sum_{n>=0} (n + y)^(2*n) * x^n/n! ), for n>=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A318633 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

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).

A266484 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(5n) (x/N^4)^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).

A266526 a(n) = coefficient of x^ny^(n+1)/n! in Log( Sum_{n>=0} (n + y)^(2n) * x^n/n! ), for n>=1.

A319147 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N^2 + Nn + n^2)^n (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).

A266522 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2n) (x/N)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(2n) (x/N)^n/n! ]^(1/N).