cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • 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.) )

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 = 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)^(4*n) * (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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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!.

Examples

			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).

Links

Crossrefs

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

Programs

  • PARI
    /* 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.) )
  • PARI
    /* 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

Formula

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)^(3*n) * (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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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!.

Examples

			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).

Links

Crossrefs

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

Programs

  • PARI
    /* 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.) )
  • PARI
    {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

Formula

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)^(5*n) * (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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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!.

Examples

			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).

Links

Crossrefs

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

Programs

  • PARI
    /* 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.) )
  • PARI
    /* 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

Formula

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

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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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!.

Examples

			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).

Crossrefs

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

Programs

  • PARI
    /* 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.) )

A359926 a(n) = coefficient of x^n*y^(n+1)/n! in (1/4) * log( Sum_{n>=0} (n + 2*y)^(2*n) * x^n/n! ).

Original entry on oeis.org

1, 8, 168, 6016, 309760, 20957184, 1762991104, 177690607616, 20895204704256, 2810343286374400, 425698411965054976, 71735043897868419072, 13313460758336789020672, 2698754565131159025483776, 593332971403056575938560000, 140634107346363806457259884544
Offset: 1

Views

Author

Paul D. Hanna, Jan 20 2023

Keywords

Examples

			E.g.f. 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! + 20895204704256*x^9/9! + 2810343286374400*x^10/10! + ...
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! where a(n) equals the coefficient of y^(n+1)*x^n/n! in the series given by
(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! + (309760*y^6 + 2304384*y^5 + 7521360*y^4 + 13763280*y^3 + 14855385*y^2 + 8940045*y + 9342629/4)*x^5/5! + (20957184*y^7 + 200377344*y^6 + 865825536*y^5 + 2188392960*y^4 + 3486312960*y^3 + 3490688496*y^2 + 2027376336*y + 525120804)*x^6/6! + ...
Exponentiation yields the e.g.f. of A266485:
exp(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! + ... + A266485(n)*x^n/n! + ...
which equals
lim_{N->oo} [ Sum_{n>=0} (N + 2*n)^(2*n) * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 4, 56, 1504, 61952, 3492864, 251855872, 22211325952, 2321689411584, ...].

Links

Crossrefs

Cf. A266485, A359928, A319834, A318634.

Programs

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

Formula

a(n) ~ c * d^n * n! / n^(5/2), where d = 4*(1 + sqrt(2)) * exp(2 - sqrt(2)) = 17.347603772617734513447467379678826546908822081006190652539615... and c = sqrt((2 - sqrt(2))/Pi)/4 = 0.107953003168342979028946547859477378793474... - Vaclav Kotesovec, Feb 13 2023, updated Mar 17 2024

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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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!.

Examples

			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).

Crossrefs

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

Programs

  • PARI
    /* 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.) )

A359917 E.g.f.: lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).

Original entry on oeis.org

1, 1, 3, 28, 413, 9216, 268327, 9831424, 432251577, 22259307520, 1313366140331, 87431498993664, 6482838033725077, 529958491541291008, 47356678577690489295, 4592761099982656823296, 480465410003489098874993, 53933291626260492656050176, 6466413087139041540884403667
Offset: 0

Views

Author

Paul D. Hanna, Jan 21 2023

Keywords

Comments

Related limits:
(C1) exp(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + n)^n * (x/N)^n/n! ]^(1/N).
(C2) 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).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 28*x^3/3! + 413*x^4/4! + 9216*x^5/5! + 268327*x^6/6! + 9831424*x^7/7! + 432251577*x^8/8! + 22259307520*x^9/9! + 1313366140331*x^10/10! + ...
where A(x) equals the limit, as N -> oo, of the series
[1 + (N^2+N+2)*(x/N) + (N^2+2*N+2*2^2)^2*(x/N)^2/2! + (N^2+3*N+2*3^2)^3*(x/N)^3/3! + (N^2+4*N+2*4^2)^4*(x/N)^4/4! + (N^2+5*N+2*5^2)^5*(x/N)^5/5! + (N^2+6*N+2*6^2)^6*(x/N)^6/6! + ...]^(1/N).
RELATED SERIES.
The logarithm of the g.f. A(x) begins:
(a) log(A(x)) = x + 2*x^2/2! + 21*x^3/3! + 304*x^4/4! + 6985*x^5/5! + 205056*x^6/6! + 7607509*x^7/7! + ... + A359918(n)*x^n/n! + ...
where A359918(n) = [x^n*y^(n+1)/n!] (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*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) (1/2) * log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ) = (y^2 + 1/2*y + 1/2)*x + (2*y^3 + 15/2*y^2 + 7*y + 15/2)*x^2/2! + (21*y^4 + 197/2*y^3 + 543/2*y^2 + 639/2*y + 683/2)*x^3/3! + (304*y^5 + 2495*y^4 + 8984*y^3 + 22246*y^2 + 29360*y + 31019)*x^4/4! + (6985*y^6 + 150489/2*y^5 + 817005/2*y^4 + 1335885*y^3 + 3162830*y^2 + 8940045/2*y + 9342629/2)*x^5/5! + (205056*y^7 + 2946228*y^6 + 20587128*y^5 + 94146240*y^4 + 294518400*y^3 + 684700836*y^2 + 1013688168*y + 1050241608)*x^6/6! + ...

Links

Crossrefs

Cf. A359918, A266485, A359927, A319147, A266481, A318633, A319834.

Programs

  • PARI
    /* Using formula for the logarithm of g.f. A(x) */
{L(n) = (1/2) * n! * polcoeff( polcoeff( log( sum(m=0, n+1, (m^2 + m*y + 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), ", "))
  • PARI
    /* Using limit formula */
\p100
P(n) = sum(k=0, 31, (n^2 + n*k + 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.) )

Formula

E.g.f. A(x) = Sum_{n>=0} a(n) * x^n/n! may be defined by the following.
(1) A(x) = lim_{N->oo} [ Sum_{n>=0} (N^2 + N*n + 2*n^2)^n * (x/N)^n/n! ]^(1/N).
(2) A(x) = exp( Sum_{n>=0} A359918(n)*x^n/n! ), where A359918(n) = (1/2) * [x^n*y^(n+1)/n!] log( Sum_{n>=0} (n^2 + n*y + 2*y^2)^n * x^n/n! ).
a(n) ~ c * d^n * n! / n^(5/2), where d = 7.68892218919697462312... and c = 0.155267010681833... - Vaclav Kotesovec, Mar 21 2024

A266488 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(2*n) * (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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

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!.

Examples

			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; ...

Crossrefs

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

Programs

  • 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))))
Showing 1-9 of 9 results.

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