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.

A051232 9-factorial numbers.

Original entry on oeis.org

1, 9, 162, 4374, 157464, 7085880, 382637520, 24106163760, 1735643790720, 140587147048320, 12652843234348800, 1252631480200531200, 135284199861657369600, 15828251383813912243200, 1994359674360552942643200, 269238556038674647256832000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_9)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
a(n) = 9*A035023(n) = Product_{k=1..n} 9*k, n >= 1; a(0) := 1.
Pi^n/a(n) is the volume of a 2n-dimensional ball with radius 1/3. - Peter Luschny, Jul 24 2012

Links

Crossrefs

Cf. A047058, A051188, A051189. a(n) = A051231(n-1, 0), A053116 (first column of triangle).

Programs

  • Magma
    [9^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011
  • Maple
    with(combstruct):A:=[N,{N=Cycle(Union(Z$9))},labeled]: seq(count(A,size=n+1)/9, n=0..14); # Zerinvary Lajos, Dec 05 2007
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 8, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

Formula

a(n) = n!*9^n =: (9*n)(!^9).
E.g.f.: 1/(1-9*x).
G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 18*x/(1 - 18*x/(1 - 27*x/(1 - 27*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, Aug 09 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/9).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/9). (End)

A147629 9-factorial numbers (4).

Original entry on oeis.org

1, 5, 70, 1610, 51520, 2112320, 105616000, 6231344000, 423731392000, 32627317184000, 2805949277824000, 266565181393280000, 27722778864901120000, 3132674011733826560000, 382186229431526840320000, 50066396055530016081920000, 7009295447774202251468800000
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Keywords

Links

Crossrefs

Cf. A035012, A035013, A035017, A035018, A035020, A035022, A035023, A045756, A048994, A049211, A051232, A053116, A132393.

Programs

  • Magma
    [Round(9^(n-1)*Gamma(n-1 +5/9)/Gamma(5/9)): n in [1..20]]; // G. C. Greubel, Dec 03 2019
  • Maple
    seq(9^(n-1)*pochhammer(5/9, n-1), n = 1..20); # G. C. Greubel, Dec 03 2019
  • Mathematica
    Table[9^(n-1)*Pochhammer[5/9, n-1], {n,20}] (* G. C. Greubel, Dec 03 2019 *)
  • PARI
    vector(20, n, prod(j=0,n-2, 9*j+5) ) \\ G. C. Greubel, Dec 03 2019
  • Sage
    [9^(n-1)*rising_factorial(5/9, n-1) for n in (1..20)] # G. C. Greubel, Dec 03 2019

Formula

a(n+1) = Sum_{k=0..n} A132393(n,k)*5^k*9^(n-k). - Philippe Deléham, Nov 09 2008
From R. J. Mathar, Nov 09 2008: (Start)
a(n) = a(n-1) + (4 + 9*(n-2))*a(n-1) = (9*n-13)*a(n-1).
a(n) = 9^(n-1)*Gamma(n-4/9)/Gamma(5/9).
G.f.: z*2F0(5/9,1; -; 9*z). (End)
a(n) = (-4)^n*Sum_{k=0..n} (9/4)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). - Amiram Eldar, Dec 21 2022

A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

Original entry on oeis.org

1, 6, 90, 2160, 71280, 2993760, 152681760, 9160905600, 632102486400, 49303993939200, 4289447472710400, 411786957380198400, 43237630524920832000, 4929089879840974848000, 606278055220439906304000, 80028703289098067632128000, 11284047163762827536130048000
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Keywords

Comments

Original name was: 9-factorial numbers (5).

Crossrefs

Cf. A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.
Cf. A048994, A097188, A132393.

Programs

  • Magma
    [Round(9^n*Gamma(n+6/9)/Gamma(6/9)): n in [0..20]]; // Vincenzo Librandi, Feb 21 2015
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,9}];lst
Table[Product[9k-3,{k,1,n-1}],{n,20}] (* Harvey P. Dale, Sep 01 2016 *)
  • Maxima
    a(n):=n!*sum(binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1),k,1,n-1)/n; /* Vladimir Kruchinin, Apr 01 2011 */
  • PARI
    a(n) = n--; prod(k=1, n, 9*k-3); \\ Michel Marcus, Feb 28 2015

Formula

a(n+1) = Sum_{k, 0<=k<=n}A132393(n,k)*6^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = n!*(Sum_{k=1..n-1} binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1))/n for n>1, also a(n) = n!*A097188(n-1). - Vladimir Kruchinin, Apr 01 2011
a(n) = (-3)^n*sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = round(9^n * Gamma(n+6/9) / Gamma(6/9)). - Vincenzo Librandi, Feb 21 2015
Sum_{n>=1} 1/a(n) = 1 + (e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). - Amiram Eldar, Dec 21 2022

Extensions

New name from Peter Bala, Feb 20 2015
More terms from Michel Marcus, Feb 28 2015

A147631 9-factorial numbers (6).

Original entry on oeis.org

1, 7, 112, 2800, 95200, 4093600, 212867200, 12984899200, 908942944000, 71806492576000, 6318971346688000, 612940220628736000, 64971663386646016000, 7471741289464291840000, 926495919893572188160000, 123223957345845101025280000, 17497801943110004345589760000
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Keywords

Crossrefs

Cf. A147630, A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.
Cf. A048994, A132393.

Programs

  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,6,2*5!,9}];lst

Formula

a(n+1) = Sum_{k=0..n} A132393(n,k)*7^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = (-2)^n*Sum_{k=0..n} (9/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
Sum_{n>=1} 1/a(n) = 1 + (e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). - Amiram Eldar, Dec 21 2022
Showing 1-4 of 4 results.

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