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 1-4 of 4 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

A266523 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N).

Original entry on oeis.org

1, 3, 54, 1737, 80460, 4866075, 363195144, 32252007249, 3320837109648, 388974074329395, 51071746190248800, 7429243977263853657, 1185973466659967427264, 206128694834273499148107, 38747184998101320725389440, 7832602778214436587234950625, 1694328566956587966290832896256, 390523839870137752804243701312099, 95545779571238219801892087161845248, 24730355203857044123269648640967753705, 6751503716745494652518864431722119040000, 1938877409334089151858199776112230794503803
Offset: 0

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

Comments

The e.g.f. A(x) of this sequence also satisfies:
A(x*y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(3*n) * (x/N^2)^n/n! ] / G(x,y)^N
where
G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(3*n) * (x/N^2)^n/n! ]^(1/N)
for all real y.

Examples

			E.g.f.: A(x) = 1 + 3*x + 54*x^2/2! + 1737*x^3/3! + 80460*x^4/4! + 4866075*x^5/5! + 363195144*x^6/6! + 32252007249*x^7/7! + 3320837109648*x^8/8! + 388974074329395*x^9/9! + 51071746190248800*x^10/10! +...
such that
A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ] / F(x)^N
where
F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(3*n) * (x/N^2)^n/n! ]^(1/N)
and
F(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! +...+ A266482(n)*x^n/n! +...

Crossrefs

Cf. A266482, A266522, A266524, A266525.

A266524 E.g.f.: Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ] / F(x)^N, where F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N).

Original entry on oeis.org

1, 4, 100, 4464, 286816, 24053120, 2488967136, 306383969920, 43726697867008, 7098711727021056, 1291743506952832000, 260410631081389420544, 57609344863582419640320, 13875489289115958335143936, 3614364399291754755286614016, 1012444950785630853817442304000, 303479487751656117544078504493056, 96925825525767333731669511270563840, 32859305845564004294368688506268024832, 11784943093649049136596829229809817092096, 4458038385946160559288726139220234076160000, 1773928724624151210275576625473634276174987264, 740706616375525604793089813921394696991733186560
Offset: 0

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

Comments

The e.g.f. A(x) of this sequence also satisfies:
A(x*y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(4*n) * (x/N^3)^n/n! ] / G(x,y)^N
where
G(x,y) = Limit_{N->oo} [ Sum_{n>=0} (N + n*y)^(4*n) * (x/N^3)^n/n! ]^(1/N)
for all real y.

Examples

			E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 4464*x^3/3! + 286816*x^4/4! + 24053120*x^5/5! + 2488967136*x^6/6! + 306383969920*x^7/7! + 43726697867008*x^8/8! + 7098711727021056*x^9/9! + 1291743506952832000*x^10/10! +...
such that
A(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ] / F(x)^N
where
F(x) = Limit_{N->oo} [ Sum_{n>=0} (N + n)^(4*n) * (x/N^3)^n/n! ]^(1/N)
and
F(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! +...+ A266483(n)*x^n/n! +...

Crossrefs

Cf. A266483, A266522, A266523, A266525.

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).

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

Views

Author

Paul D. Hanna, Dec 30 2015

Keywords

Comments

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.

Examples

			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! +...

Crossrefs

Cf. A266484, A266522, A266523, A266524.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.