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 1-6 of 6 results.

A094856 E.g.f.: exp(4x)/(1-4x)^(1/4).

Original entry on oeis.org

1, 5, 29, 217, 2297, 34349, 674965, 16276481, 461527793, 14993138773, 548258687501, 22272738733865, 994870668959209, 48451779617935997, 2554818339078836357, 144990720049391354449, 8811240401831517313505, 570857963393730507892901, 39275973938444154366908413
Offset: 0

Views

Author

Philippe Deléham, Jun 13 2004

Keywords

Comments

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n) for x = 1, 2, 3 respectively.

Links

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[E^(4x)/(1-4x)^(1/4),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 03 2012 *)
With[{nn=20},CoefficientList[Series[Exp[4x]/(1-4x)^(1/4),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Mar 29 2013 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(4*x)/(1-4*x)^(1/4))) \\ Joerg Arndt, May 11 2013

Formula

a(n) = Sum_{k = 0..n} A046716(n, k)*4^k.
a(n) ~ n^(n-1/4)*4^n*Gamma(3/4)/(exp(n-1)*sqrt(Pi)). - Vaclav Kotesovec, Oct 03 2012
Conjectured to be D-finite with recurrence: a(n) +(-4*n-1)*a(n-1) +16*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019

Extensions

Corrected and extended by Harvey P. Dale, Mar 29 2013

A094869 E.g.f.: exp(5x)/(1-5x)^(1/5).

Original entry on oeis.org

1, 6, 41, 356, 4401, 78826, 1893481, 56341416, 1978638881, 79749105326, 3622010623401, 182895318578956, 10160561511881041, 615728464210461906, 40414538467581457001, 2855999961062529064976, 216180544920721807887681
Offset: 0

Views

Author

Philippe Deléham, Jun 16 2004

Keywords

Comments

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n) for x = 1, 2, 3, 4 respectively.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[5x]/(1-5x)^(1/5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2014 *)

Formula

a(n) = Sum_{k = 0..n} A046716(n, k)*5^k.
Conjectured to be D-finite with recurrence: a(n) +(-5*n-1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * 5^n * n^(n - 3/10) / (Gamma(1/5) * exp(n-1)). - Vaclav Kotesovec, Nov 19 2021

A094905 Expansion of e.g.f.: exp(6*x)/(1-6*x)^(1/6).

Original entry on oeis.org

1, 7, 55, 541, 7585, 157231, 4452247, 157484725, 6594785281, 317357589655, 17222102537911, 1039632137764237, 69073193451776545, 5007661199176196671, 393324947394545293975, 33268708968518818629541
Offset: 0

Views

Author

Philippe Deléham, Jun 16 2004

Keywords

Comments

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n) for x = 1, 2, 3, 4, 5 respectively.

Crossrefs

Cf. A046716.
Cf. A000522, A081367, A094822, A094856, A094869.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[6x]/Surd[1-6x,6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 15 2022 *)

Formula

E.g.f.: exp(6*x)/(1-6*x)^(1/6).
a(n) = Sum_{k = 0..n} A046716(n, k)*6^k.
Conjectured to be D-finite with recurrence: a(n) +(-6*n-1)*a(n-1) +36*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(Pi) * 2^(n + 1/2) * 3^n * n^(n - 1/3) / (Gamma(1/6) * exp(n - 1)). - Vaclav Kotesovec, Nov 19 2021

A094911 Expansion of e.g.f. exp(7*x)/(1-7*x)^(1/7).

Original entry on oeis.org

1, 8, 71, 778, 12125, 284012, 9241891, 378595022, 18409947641, 1029827400400, 64998958518719, 4565303338264082, 353016345110857429, 29793105387299603252, 2724646021507044539675, 268374407984059193374678
Offset: 0

Views

Author

Philippe Deléham, Jun 17 2004

Keywords

Comments

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n) for x = 1, 2, 3, 4, 5, 6 respectively.

Crossrefs

Cf. A046716, A000522, A081367, A094822, A094856, A094869, A094905.

Programs

  • PARI
    my(x='x+O('x^20)); Vec(serlaplace(exp(7*x)/(1-7*x)^(1/7))) \\ Michel Marcus, Jan 23 2023

Formula

a(n) = Sum_{k = 0..n} A046716(n, k)*7^k.
Conjectured to be D-finite with recurrence: a(n) +(-7*n-1)*a(n-1) +49*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 7^n / (Gamma(1/7) * exp(n-1) * n^(6/7)). - Vaclav Kotesovec, Nov 19 2021

Extensions

Corrected by D. S. McNeil, Aug 20 2010

A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

Original entry on oeis.org

1, 9, 89, 1073, 18321, 476473, 17484457, 813648417, 45054110369, 2872362067433, 206710159889529, 16558892507010961, 1460688620617834801, 140655075719488236057, 14678730623948132120009
Offset: 0

Views

Author

Philippe Deléham, Jun 18 2004

Keywords

Comments

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[8x]/Surd[1-8x,8],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 25 2019 *)

Formula

a(n) = Sum_{k = 0..n} A046716(n, k)*8^k.
D-finite with recurrence: a(n) +(-8*n-1)*a(n-1) +64*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 8^n / (Gamma(1/8) * exp(n-1) * n^(7/8)). - Vaclav Kotesovec, Nov 19 2021

A095176 E.g.f.: exp(9x)/(1-9x)^(1/9).

Original entry on oeis.org

1, 10, 109, 1432, 26497, 754894, 30787885, 1603546156, 99602138593, 7128277455538, 576063289419661, 51832424202980320, 5136461847251936929, 555721381650431686582, 65167921144448534609677
Offset: 0

Views

Author

Philippe Deléham, Jun 20 2004

Keywords

Comments

Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n), A094935(n) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.
From Vaclav Kotesovec, Nov 19 2021: (Start)
In general, for k > 0, if e.g.f. = exp(k*x) / (1 - k*x)^(1/k), then a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * k^n / (Gamma(1/k) * exp(n-1) * n^(1 - 1/k)).
Equivalently, a(n) ~ n! * exp(1) * k^n / (Gamma(1/k) * n^(1 - 1/k)). (End)

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[9x]/Surd[1-9x,9],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 25 2020 *)

Formula

a(n) = Sum_{k = 0..n} A046716(n, k)*9^k.
D-finite with recurrence a(n) +(-9*n-1)*a(n-1) +81*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 9^n / (Gamma(1/9) * exp(n-1) * n^(8/9)). - Vaclav Kotesovec, Nov 19 2021
Showing 1-6 of 6 results.

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