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 1-5 of 5 results.

A034911 One fifth of octo-factorial numbers.

Original entry on oeis.org

1, 13, 273, 7917, 292929, 13181805, 698635665, 42616775565, 2940557513985, 226422928576845, 19245948929031825, 1789873250399959725, 180777198290395932225, 19704714613653156612525, 2305451609797419323665425, 288181451224677415458178125, 38328133012882096255937690625
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034908, A034909, A034910, A034911, A034912, A045755, A147625.

Programs

  • Magma
    [n le 1 select 1 else (8*n-3)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 20 2022
  • Mathematica
    With[{nn=20},CoefficientList[Series[(-1+(1-8x)^(-5/8))/5,{x,0,nn}],x] Range[0,nn]!] (* or *) FoldList[Times,Range[5,200,8]]/5 (* Harvey P. Dale, May 25 2016 *)
  • SageMath
    [8^n*rising_factorial(5/8,n)/5 for n in range(1,40)] # G. C. Greubel, Oct 20 2022

Formula

5*a(n) = (8*n-3)(!^8) = Product_{j=1..n} 8*j-3.
E.g.f.: (-1+(1-8*x)^(-5/8))/5.
G.f.: x/(1-13*x/(1-8*x/(1-21*x/(1-16*x/(1-29*x/(1-24*x/(1-37*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
D-finite with recurrence: a(n) = (8*n-3)*a(n-1). - R. J. Mathar, Jan 28 2020
a(n) = (1/5)* 8^n * Pochhammer(n, 5/8). - G. C. Greubel, Oct 20 2022
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147625(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/8^3)^(1/8)*(Gamma(5/8) - Gamma(5/8, 1/8)). (End)

A147626 Octo-factorial numbers (5).

Original entry on oeis.org

1, 6, 84, 1848, 55440, 2106720, 96909120, 5233092480, 324451733760, 22711621363200, 1771506466329600, 152349556104345600, 14320858273808486400, 1460727543928465612800, 160680029832131217408000, 18960243520191483654144000, 2388990683544126940422144000
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Keywords

Links

Crossrefs

Cf. A034908, A045755, A048994, A049210, A051189, A052714, A132393, A147625.

Programs

  • Magma
    [n le 1 select 1 else (8*n-10)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,8}];lst
Table[8^(n-1)*Pochhammer[3/4, n-1], {n,40}] (* G. C. Greubel, Oct 21 2022 *)
  • SageMath
    [8^(n-1)*rising_factorial(3/4, n-1) for n in range(1,40)] # G. C. Greubel, Oct 21 2022

Formula

a(n+1) = Sum_{k=0..n} A132393(n,k)*6^k*8^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = (-2)^n*Sum_{k=0..n} 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
G.f.: 2*x/G(0), where G(k) = 1 + 1/(1 - 2*x*(8*k+6)/(2*x*(8*k+6) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
From G. C. Greubel, Oct 21 2022: (Start)
a(n) = 8^n * Pochhammer(n, 3/4) = -2^(3*n-1) * Pochhammer(n, -1/4).
a(n) = (8*n - 10)*a(n-1). (End)
Sum_{n>=1} 1/a(n) = 1 + (e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). - Amiram Eldar, Dec 20 2022

A153271 Triangle T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3, read by rows.

Original entry on oeis.org

5, 5, 30, 5, 35, 315, 5, 40, 440, 6160, 5, 45, 585, 9945, 208845, 5, 50, 750, 15000, 375000, 11250000, 5, 55, 935, 21505, 623645, 21827575, 894930575, 5, 60, 1140, 29640, 978120, 39124800, 1838865600, 99298742400, 5, 65, 1365, 39585, 1464645, 65909025, 3493178325, 213083877825, 14702787569925
Offset: 0

Views

Author

Roger L. Bagula, Dec 22 2008

Keywords

Comments

Row sums are {5, 35, 355, 6645, 219425, 11640805, 917404295, 101177741765, 14919432040765, 2839006665525525, 677815000136926955, ...}.

Examples

			Triangle begins as:
  5;
  5, 30;
  5, 35, 315;
  5, 40, 440,  6160;
  5, 45, 585,  9945, 208845;
  5, 50, 750, 15000, 375000, 11250000;
  5, 55, 935, 21505, 623645, 21827575, 894930575;

Links

Crossrefs

Cf. A153271 (m=2), this sequence (m=3), A153272 (m=4).
Sequences related to m values:
m=2: A001147, A001710, A008545, A032031, A047056, A144739, A144758.
m=3: A001720, A007696, A008543, A034000, A051577, A052562, A147585, A147625, A147629.

Programs

  • Magma
    m:=3;
function T(n,k)
  if k eq 0 then return NthPrime(m);
  else return (&*[j*n + NthPrime(m): j in [0..k]]);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 03 2019
  • Maple
    m:=3; seq(seq(`if`(k=0, ithprime(m), mul(j*n + ithprime(m), j=0..k)), k=0..n), n=0..10); # G. C. Greubel, Dec 03 2019
  • Mathematica
    T[n_, k_, m_]:= If[k==0, Prime[m], Product[j*n + Prime[m], {j,0,k}]];
Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten
  • PARI
    T(n,k) = my(m=3); if(k==0, prime(m), prod(j=0,k, j*n + prime(m)) ); \\ G. C. Greubel, Dec 03 2019
  • Sage
    def T(n, k):
    m=3
    if (k==0): return nth_prime(m)
    else: return product(j*n + nth_prime(m) for j in (0..k))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 03 2019

Formula

T(n, k) = Product_{j=0..k} (j*n + prime(m)), with T(n, 0) = prime(m) and m = 3.

Extensions

Edited by G. C. Greubel, Dec 03 2019

A268730 a(n) = Product_{k = 0..n} 2*(8*k + 5).

Original entry on oeis.org

10, 260, 10920, 633360, 46868640, 4218177600, 447126825600, 54549472723200, 7527827235801600, 1159285394313446400, 197078517033285888000, 36656604168191175168000, 7404634041974617383936000, 1614210221150466589698048000, 377725191749209181989343232000
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 12 2016

Keywords

Examples

			a(0) = (1 + 2 + 3 + 4) = 10;
a(1) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) = 260;
a(2) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12) = 10920;
a(3) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12)*(13 + 14 + 15 + 16) = 633360, etc.

Links

Crossrefs

Cf. A000027, A113770, A147630.

Programs

  • Magma
    [&*[(16*k+10): k in [0..n-1]]: n in [1..20]]; // Vincenzo Librandi, Feb 12 2016
  • Mathematica
    FullSimplify[Table[(2^(4 n + 13/4) Gamma[1/8] Gamma[n + 13/8])/(Sqrt[Pi] Gamma[1/4]), {n, 0, 14}]]
Table[Product[16 k + 10, {k, 0, n - 1}], {n, 20}] (* Vincenzo Librandi, Feb 12 2016 *)
  • PARI
    x='x+O('x^50); Vec(serlaplace(10/(1 - 16*x)^(13/8))) \\ G. C. Greubel, Apr 09 2017

Formula

a(n) = (2^(4*n + 13/4)*Gamma(1/8)*Gamma(n + 13/8))/(sqrt(Pi)*Gamma(1/4)), where Gamma(x) is the gamma function.
a(n) = 2*(8*n + 5)*a(n - 1), a(0)=10.
Sum_{n>=0} 1/a(n) = (exp(1/16)*(Gamma(5/8) - Gamma(5/8, 1/16)))/(2*sqrt(2)) = 0.10393932939417..., where Gamma(a, x) is the incomplete gamma function.
a(n) ~ sqrt(Pi) * 2^(4*n+9/2) * n^(n+9/8) / (Gamma(5/8) * exp(n)). - Vaclav Kotesovec, Feb 20 2016
G.f.: 10/(1-b(1)x/(1-(b(1)-10)x/(1-b(2)x/(1-(b(2)-10)x/(1-b(3)x/(...)))))), where b(n)=2(5+8n), i.e. 26,42,58,74. - Benedict W. J. Irwin, Feb 24 2016
a(n) = 2^(n+1)*A147625(n+2). - R. J. Mathar, Jun 07 2016
E.g.f.: 10/(1 - 16*x)^(13/8). - Ilya Gutkovskiy, Jun 07 2016

A020026 Nearest integer to Gamma(n + 5/8)/Gamma(5/8).

Original entry on oeis.org

1, 1, 1, 3, 10, 45, 251, 1666, 12701, 109544, 1054364, 11202617, 130230419, 1644159040, 22401666926, 327624378798, 5119130918712, 85105551523595, 1499985345603354, 27937227061862467, 548268081089050918
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Gamma(n + 5/8)/Gamma(5/8) = 1, 5/8, 65/64, 1365/512, 39585/4096, 1464645/32768, 65909025/262144, ... - R. J. Mathar, Sep 04 2016

Crossrefs

Cf. A020071, A020116, A147625.

Programs

  • Maple
    Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;
Showing 1-5 of 5 results.

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

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