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 1-3 of 3 results.

A045757 10-factorial numbers.

Original entry on oeis.org

1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.
Cf. A008542, A048176.

Programs

  • GAP
    List([1..20], n-> Product([0..n-1], j-> 10*j+1) ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    G(x):=-1+(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..14); # Zerinvary Lajos, Apr 03 2009
seq(mul(10*j+1, j = 0..n-1), n = 1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    FoldList[Times,10*Range[0,20]+1] (* Harvey P. Dale, Dec 02 2016 *)
  • PARI
    vector(21, n, prod(j=0,n-1, 10*j+1) ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j+1) for j in (0..n-1)) for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

a(n) = Pochhammer(1/10,n)*10^n.
a(n+1) = (10*n+1)(!^10) = Product_{k=0..n} (10*k+1), n >= 0.
E.g.f.: -1 + (1-10*x)^(-1/10).
Sum_{n>=1} 1/a(n) = (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022

A049212 a(n) = -Product_{k=0..n} (10*k - 1); deca-factorial numbers.

Original entry on oeis.org

1, 9, 171, 4959, 193401, 9476649, 559122291, 38579438079, 3047775608241, 271252029133449, 26853950884211451, 2927080646379048159, 348322596919106730921, 44933615002564768288809, 6245772485356502792144451, 930620100318118916029523199, 147968595950580907648694188641
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008543, A035278, A048994, A049209, A049210, A049211, A048176.

Programs

  • Magma
    [Round(10^n*Gamma(n+9/10)/Gamma(9/10)): n in [0..25]]; // G. C. Greubel, Feb 03 2022
  • Mathematica
    CoefficientList[Series[(1-10*x)^(-9/10),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
  • PARI
    a(n) = {-prod(k=0, n, 10*k-1)} \\ Andrew Howroyd, Jan 02 2020
  • Sage
    [10^n*rising_factorial(9/10, n) for n in (0..25)] # G. C. Greubel, Feb 03 2022

Formula

a(n) = 9*A035278(n) = (10*n-1)(!^10), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 10^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) = 10^n * Gamma(n+9/10) / Gamma(9/10). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-10*x)^(-9/10). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (20*k+9)*x - 10*(k+1)*(10*k+9)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-9*x-90*x^2/(1-29*x-380*x^2/(1-49*x-870*x^2/(1-69*x-1560*x^2/(1-89*x-2450*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(10*k+9)/(1 - x*(10*k+10)/Q(k+1)) (continued fraction).
G.f.: 1/(1-9*x/(1-10*x/(1-19*x/(1-20*x/(1-29*x/(1-30*x/(1-39*x/(1-40*x/(1-49*x/(1-50*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: Hypergeometric2F0([1, 9/10], --; 10*x). - G. C. Greubel, Feb 03 2022
Sum_{n>=0} 1/a(n) = 1 + (e/10)^(1/10)*(Gamma(9/10) - Gamma(9/10, 1/10)). - Amiram Eldar, Dec 22 2022

Extensions

Terms a(14) and beyond from Andrew Howroyd, Jan 02 2020

A051262 10-factorial numbers.

Original entry on oeis.org

1, 10, 200, 6000, 240000, 12000000, 720000000, 50400000000, 4032000000000, 362880000000000, 36288000000000000, 3991680000000000000, 479001600000000000000, 62270208000000000000000
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_10)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Links

Crossrefs

a(n) = A048176(n+1, 0)*(-1)^n (first column of unsigned triangle).
Cf. A047058, A051188, A051189, A051232, A035279.

Programs

  • Magma
    [10^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 05 2011
  • Maple
    with(combstruct):A:=[N,{N=Cycle(Union(Z$10))},labeled]: seq(count(A,size=n)/10,n=0..14); # Zerinvary Lajos, Dec 05 2007
  • Mathematica
    Array[#!*10^# &, 14, 0] (* Michael De Vlieger, Sep 04 2017 *)

Formula

a(n) = 10*A035279(n) = Product_{k=1..n} 10*k, n >= 1; a(0) := 1.
a(n) = n!*10^n =: (10*n)(!^10);
E.g.f.: 1/(1-10*x).
G.f.: 1/(1 - 10*x/(1 - 10*x/(1 - 20*x/(1 - 20*x/(1 - 30*x/(1 - 30*x/(1 - ...))))))), a continued fraction. - Ilya Gutkovskiy, May 12 2017
From Amiram Eldar, Jun 25 2020: (Start)
Sum_{n>=0} 1/a(n) = e^(1/10).
Sum_{n>=0} (-1)^n/a(n) = e^(-1/10). (End)
Showing 1-3 of 3 results.

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