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 1-10 of 13 results. Next

A049211 a(n) = Product_{k=1..n} (9*k - 1); 9-factorial numbers.

Original entry on oeis.org

1, 8, 136, 3536, 123760, 5445440, 288608320, 17893715840, 1270453824640, 101636305971200, 9045631231436800, 886471860680806400, 94852489092846284800, 11002888734770169036800, 1375361091846271129600000, 184298386307400331366400000, 26354669241958247385395200000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), A049210 (m=8), this sequence (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).
Cf. A035022, A048994.

Programs

  • Magma
    m:=9; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..20]]; // G. C. Greubel, Feb 08 2022
  • Mathematica
    CoefficientList[Series[(1-9*x)^(-8/9),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
  • PARI
    a(n) = prod(k=1, n, 9*k-1); \\ Michel Marcus, Jan 08 2015
  • Sage
    m=9; [m^n*rising_factorial((m-1)/m, n) for n in (0..20)] # G. C. Greubel, Feb 08 2022

Formula

a(n) = 8*A035022(n) = (9*n-1)(!^9), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 9^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) = 9^n * Gamma(n+8/9) / Gamma(8/9). - Vaclav Kotesovec, Jan 28 2015
E.g.f: (1-9*x)^(-8/9). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Dec 09 2020: (Start)
G.f.: 1/(1-8*x-72*x^2/(1-26*x-306*x^2/(1-44*x-702*x^2/(1-62*x-1260*x^2/(1-80*x-1980*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/(1-8*x/(1-9*x/(1-17*x/(1-18*x/(1-26*x/(1-27*x/(1-35*x/(1-36*x/(1-44*x/(1-45*x/(1-...))))))))))) (Stieltjes continued fraction). (End)
From Nikolaos Pantelidis, Dec 19 2020: (Start)
G.f.: 1/G(0) where G(k) = 1 - (18*k+8)*x - 9*(k+1)*(9*k+8)*x^2/G(k+1) (continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(9*k+8)/(1 - x*(9*k+9)/Q(k+1) ) (continued fraction). (End)
G.f.: hypergeometric2F0([1, 8/9], [--], 9*x). - G. C. Greubel, Feb 08 2022
Sum_{n>=0} 1/a(n) = 1 + (e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 1/9)). - Amiram Eldar, Dec 21 2022

Extensions

a(9) (originally given incorrectly as 1011636305971200) corrected by Peter Bala, Feb 20 2015
a(15)-a(16) from Vincenzo Librandi, Feb 20 2015
a(16) corrected and incorrect MAGMA program removed by Georg Fischer, May 10 2021

A035012 One half of 9-factorial numbers.

Original entry on oeis.org

1, 11, 220, 6380, 242440, 11394680, 638102080, 41476635200, 3069271004800, 254749493398400, 23436953392652800, 2367132292657932800, 260384552192372608000, 30985761710892340352000, 3966177498994219565056000, 543366317362208080412672000, 79331482334882379740250112000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007558, A034171, A035012, A035013, A035017, A035018.
Cf. A035020, A035021, A035022, A035023, A045756, A084949.

Programs

  • Magma
    [n le 1 select 1 else (9*n-7)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 10, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[2/9, n]/2, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)
  • SageMath
    [9^n*rising_factorial(2/9,n)/2 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

Formula

2*a(n) = (9*n-7)(!^9) := Product_{j=1..n} (9*j - 7).
E.g.f.: (-1+(1-9*x)^(-2/9))/2.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/2) * 9^n * Pochhammer(n, 2/9).
a(n) = (9*n-7)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A084949(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). (End)

A035023 One ninth of 9-factorial numbers.

Original entry on oeis.org

1, 18, 486, 17496, 787320, 42515280, 2678462640, 192849310080, 15620794116480, 1405871470483200, 139181275577836800, 15031577762406374400, 1758694598201545804800, 221595519373394771404800, 29915395115408294139648000, 4307816896618794356109312000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

E.g.f. is g.f. for A001019(n-1) (powers of nine).

Links

Crossrefs

Cf. A000142, A007559, A001019, A034171, A035012, A035013.
Cf. A035017, A035018, A035020, A035021, A035022, A045756.

Programs

  • Magma
    [9^(n-1)*Factorial(n): n in [1..40]]; // G. C. Greubel, Oct 19 2022
  • Mathematica
    With[{nn=20},Rest[CoefficientList[Series[(-1+1/(1-9*x))/9,{x,0,nn}],x] Range[ 0,nn]!]] (* Harvey P. Dale, Apr 07 2019 *)
Table[9^(n-1)*n!, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)
  • SageMath
    [9^(n-1)*factorial(n) for n in range(1,40)] # G. C. Greubel, Oct 19 2022

Formula

9*a(n) = (9*n)(!^9) = Product_{j=1..n} 9*j = 9^n*n!.
E.g.f.: (-1+1/(1-9*x))/9.
D-finite with recurrence: a(n) - 9*n*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 9*(exp(1/9)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*(1-exp(-1/9)). (End)
a(n) = A001019(n-1) * A000142(n). - G. C. Greubel, Oct 19 2022

A035013 One third of 9-factorial numbers.

Original entry on oeis.org

1, 12, 252, 7560, 294840, 14152320, 806682240, 53241027840, 3993077088000, 335418475392000, 31193918211456000, 3181779657568512000, 353177541990104832000, 42381305038812579840000, 5467188350006822799360000, 754471992300941546311680000, 110907382868238407307816960000, 17301551727445191540019445760000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

E.g.f. is g.f. for A034171(n-1).

Links

Crossrefs

Cf. A007559, A034171, A035012, A035013, A035017, A035018.
Cf. A035020, A035021, A035022, A035023, A045756, A144758.

Programs

  • Magma
    [n le 1 select 1 else (9*n-6)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 11, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[1/3, n]/3, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)
  • SageMath
    [9^n*rising_factorial(1/3,n)/3 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

Formula

3*a(n) = (9*n-6)(!^9) := Product_{j=1..n} (9*j-6) = 3^n*A007559(n).
E.g.f.: (-1+(1-9*x)^(-1/3))/3.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/3) * 9^n * Pochhammer(n, 1/3).
a(n) = (9*n-6)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A144758(n)/3.
Sum_{n>=1} 1/a(n) = 3*(e/9^6)^(1/9)*(Gamma(1/3) - Gamma(1/3, 1/9)). (End)

Extensions

Terms a(15) onward added by G. C. Greubel, Oct 18 2022

A035017 One quarter of 9-factorial numbers.

Original entry on oeis.org

1, 13, 286, 8866, 354640, 17377360, 1007886880, 67528420960, 5132159992960, 436233599401600, 41005958343750400, 4223613709406291200, 473044735453504614400, 57238412989874058342400, 7440993688683627584512000, 1034298122727024234247168000, 153076122163599586668580864000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A034171, A035012, A035013, A035017, A035018.
Cf. A035020, A035021, A035022, A035023, A045756, A144829.

Programs

  • Magma
    [n le 1 select 1 else (9*n-5)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[4/9, n]/4, {n,40}] (* G. C. Greubel, Oct 18 2022 *)
  • SageMath
    [9^n*rising_factorial(4/9,n)/4 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

Formula

4*a(n) = (9*n-5)(!^9) := Product_{j=1..n} (9*j-5).
E.g.f.: (-1+(1-9*x)^(-4/9))/4.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/4) * 9^n * Pochhammer(n, 4/9).
a(n) = (9*n-5)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A144829(n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). (End)

A035018 One fifth of 9-factorial numbers.

Original entry on oeis.org

1, 14, 322, 10304, 422464, 21123200, 1246268800, 84746278400, 6525463436800, 561189855564800, 53313036278656000, 5544555772980224000, 626534802346765312000, 76437245886305368064000, 10013279211106003216384000, 1401859089554840450293760000, 208877004343671227093770240000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A034171, A035012, A035013, A035017, A035018.
Cf. A035020, A035021, A035022, A035023, A045756, A147629.

Programs

  • Magma
    [n le 1 select 1 else (9*n-4)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Rest[FoldList[Times,1,9*Range[20]-4]/5] (* Harvey P. Dale, May 22 2013 *)
  • SageMath
    [9^n*rising_factorial(5/9,n)/5 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

Formula

5*a(n) = (9*n-4)(!^9) := Product_{j=1..n} (9*j-4).
E.g.f.: (-1+(1-9*x)^(-5/9))/5.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/5) * 9^n * Pochhammer(n, 5/9).
a(n) = (9*n-4)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147629(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/9^4)^(1/9)*(Gamma(5/9) - Gamma(5/9, 1/9)). (End)

A035020 One sixth of 9-factorial numbers.

Original entry on oeis.org

1, 15, 360, 11880, 498960, 25446960, 1526817600, 105350414400, 8217332323200, 714907912118400, 68631159563366400, 7206271754153472000, 821514979973495808000, 101046342536739984384000, 13338117214849677938688000, 1880674527293804589355008000, 282101179094070688403251200000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A034000, A034171, A035012, A035013, A035017, A147630.
Cf. A035018, A035020, A035021, A035022, A035023, A045756.

Programs

  • Magma
    [n le 1 select 1 else (9*n-3)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 18 2022
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 2*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
Table[9^n*Pochhammer[2/3, n]/6, {n, 40}] (* G. C. Greubel, Oct 18 2022 *)
  • SageMath
    [9^n*rising_factorial(2/3,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 18 2022

Formula

6*a(n) = (9*n-3)(!^9) := Product_{j=1..n} (9*j-3) = 3^n*2*A034000(n), where 2*A034000(n) = (3*n-1)(!^3) := Product_{j=1..n} (3*j-1).
E.g.f.: (-1+(1-9*x)^(-2/3))/6.
From G. C. Greubel, Oct 18 2022: (Start)
a(n) = (1/6) * 9^n * Pochhammer(n, 2/3).
a(n) = (9*n - 3)*a(n-1). (End)
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147630(n+1)/6.
Sum_{n>=1} 1/a(n) = 6*(e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). (End)

A035021 One seventh of 9-factorial numbers.

Original entry on oeis.org

1, 16, 400, 13600, 584800, 30409600, 1854985600, 129848992000, 10258070368000, 902710192384000, 87562888661248000, 9281666198092288000, 1067391612780613120000, 132356559984796026880000, 17603422477977871575040000, 2499685991872857763655680000, 377452584772801522312007680000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A034171, A035012, A035013, A035017, A035018.
Cf. A035020, A035021, A035022, A035023, A045756, A147631.

Programs

  • Magma
    [n le 1 select 1 else (9*n-2)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022
  • Maple
    f := gfun:-rectoproc({(9*n - 2)*a(n - 1) - a(n) = 0, a(1) = 1}, a(n), remember);
map(f, [$ (1 .. 20)]); # Georg Fischer, Feb 15 2020
  • Mathematica
    Table[9^n*Pochhammer[7/9, n]/7, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)
  • SageMath
    [9^n*rising_factorial(7/9,n)/7 for n in range(1,40)] # G. C. Greubel, Oct 19 2022

Formula

7*a(n) = (9*n-2)(!^9) := Product_{j=1..n} (9*j-2).
E.g.f.: (-1+(1-9*x)^(-7/9))/7.
D-finite with recurrence: a(1) = 1, a(n) = (9*n - 2)*a(n-1) for n > 1. - Georg Fischer, Feb 15 2020
a(n) = (1/7) * 9^n * Pochhammer(n, 7/9). - G. C. Greubel, Oct 19 2022
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A147631(n+1)/7.
Sum_{n>=1} 1/a(n) = 7*(e/9^2)^(1/9)*(Gamma(7/9) - Gamma(7/9, 1/9)). (End)

Extensions

Terms a(15) onward added by G. C. Greubel, Oct 19 2022

A035024 Expansion of 1/(1-81*x)^(1/9), related to 9-factorial numbers A045756.

Original entry on oeis.org

1, 9, 405, 23085, 1454355, 96860043, 6683342967, 472607824095, 34027763334840, 2484026723443320, 183321172190117016, 13649094547609621464, 1023682091070721609800, 77248625487721376862600, 5859860019140007302005800, 446521333458468556412841960, 34158882009572844565582409940
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A034171, A035012, A035013, A035017, A035018, A035020, A035021, A035022, A035023, A045756, A256190.

Programs

  • Magma
    [n le 1 select 1 else 9*(9*n-17)*Self(n-1)/(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022
  • Mathematica
    CoefficientList[Series[1/Surd[1-81x,9],{x,0,20}],x] (* Harvey P. Dale, Mar 08 2018 *)
Table[9^(2*n)*Pochhammer[1/9, n]/n!, {n,0,40}] (* G. C. Greubel, Oct 19 2022 *)
  • SageMath
    [9^(2*n)*rising_factorial(1/9,n)/factorial(n) for n in range(40)] # G. C. Greubel, Oct 19 2022

Formula

a(n) = 9^n*A045756(n)/n!, n >= 1, where A045756(n) = (9*n-8)(!^9) = Product_{j=1..n} (9*j - 8).
G.f.: (1-81*x)^(-1/9).
D-finite with recurrence: n*a(n) = 9*(9*n-8)*a(n-1). - R. J. Mathar, Jan 28 2020
a(n) = 9^(2*n) * Pochhammer(n, 1/9)/n!. - G. C. Greubel, Oct 19 2022
a(n) ~ 3^(4*n) * n^(-8/9) / Gamma(1/9). - Amiram Eldar, Aug 18 2025

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
Showing 1-10 of 13 results. Next

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

Content is available under The OEIS End-User License Agreement.