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.

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

A266522 E.g.f.: 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).

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

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)^(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.

Examples

			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! + ...

Crossrefs

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

Formula

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

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.

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.

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