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

A035022 One eighth of 9-factorial numbers.

Original entry on oeis.org

1, 17, 442, 15470, 680680, 36076040, 2236714480, 158806728080, 12704538246400, 1130703903929600, 110808982585100800, 11856561136605785600, 1375361091846271129600, 171920136480783891200000, 23037298288425041420800000, 3294333655244780923174400000, 500738715597206700322508800000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A007559, A034000, A034171, A035012, A035013, A035017, A049211.
Cf. A035018, A035020, A035021, A035022, A035023, A045756.

Programs

  • Magma
    [n le 1 select 1 else (9*n-1)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 19 2022
  • Maple
    f := gfun:-rectoproc({(9*n - 1)*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[8/9, n]/8, {n, 40}] (* G. C. Greubel, Oct 19 2022 *)
  • SageMath
    [9^n*rising_factorial(8/9,n)/8 for n in range(1,40)] # G. C. Greubel, Oct 19 2022

Formula

8*a(n) = (9*n-1)(!^9) := Product_{j=1..n} (9*j - 1).
a(n) = (9*n)!/(n!*2^4*3^(4*n)*5*7*A045756(n)*A035012(n)*A007559(n)*A035017(n) *A035018(n)*A034000(n) *A035021(n)).
E.g.f.: (-1+(1-9*x)^(-8/9))/8.
D-finite with recurrence: a(1) = 1, a(n) = (9*n - 1)*a(n-1) for n > 1. - Georg Fischer, Feb 15 2020
a(n) = (1/8) * 9^n * Pochhammer(n, 8/9). - G. C. Greubel, Oct 19 2022
From Amiram Eldar, Dec 21 2022: (Start)
a(n) = A049211(n)/8.
Sum_{n>=1} 1/a(n) = 8*(e/9)^(1/9)*(Gamma(8/9) - Gamma(8/9, 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)

A084949 a(n) = Product_{i=0..n-1} (9*i+2).

Original entry on oeis.org

1, 2, 22, 440, 12760, 484880, 22789360, 1276204160, 82953270400, 6138542009600, 509498986796800, 46873906785305600, 4734264585315865600, 520769104384745216000, 61971523421784680704000, 7932354997988439130112000, 1086732634724416160825344000, 158662964669764759480500224000
Offset: 0

Views

Author

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Keywords

Links

Crossrefs

Cf. A000165, A008544, A001813, A047055, A047657, A084947, A084948.
Cf. A000079, A000142, A035012, A048994, A084944.

Programs

  • GAP
    List([0..20], n-> Product([0..n-1], k-> 9*k+2) ); # G. C. Greubel, Aug 19 2019
  • Magma
    [1] cat [(&*[9*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 19 2019
  • Maple
    a:= n-> product(9*i+2,i=0..n-1); seq(a(j),j=0..20);
  • Mathematica
    Table[9^n*Pochhammer[2/9, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)
  • PARI
    vector(20, n, n--; prod(k=0, n-1, 9*k+2)) \\ G. C. Greubel, Aug 19 2019
  • Sage
    [product(9*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

Formula

a(n) = A084944(n)/A000142(n)*A000079(n) = 9^n*Pochhammer(2/9, n) = 9^n*Gamma(n+2/9)/Gamma(2/9).
a(n) = (-7)^n*Sum_{k=0..n} (9/7)^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
E.g.f.: (1-9*x)^(-2/9). - Robert Israel, Mar 22 2017
D-finite with recurrence: a(n) + (-9*n+7)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/9^7)^(1/9)*(Gamma(2/9) - Gamma(2/9, 1/9)). - Amiram Eldar, Dec 21 2022
a(n) ~ sqrt(2*Pi) * (9/e)^n * n^(n-5/18) / Gamma(2/9). - Amiram Eldar, Aug 30 2025

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

A053116 a(n) = ((9*n+10)(!^9))/10, related to A045756 ((9*n+1)(!^9) 9-factorials).

Original entry on oeis.org

1, 19, 532, 19684, 905464, 49800520, 3187233280, 232668029440, 19078778414080, 1736168835681280, 173616883568128000, 18924240308925952000, 2233060356453262336000, 283598665269564316672000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Comments

Row m=10 of the array A(10; m,n) := ((9*n+m)(!^9))/m(!^9), m >= 0, n >= 0.

Links

Crossrefs

Cf. A051232, A045756, A035012-3, A035017-8, A035020-3 (rows m=0..9).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/(1 - 9*x)^(19/9))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 26 2018
  • Mathematica
    s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 18, 3*5!, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nmax = 50}, CoefficientList[Series[1/(1 - 9*x)^(19/9), {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Aug 26 2018 *)
  • PARI
    x='x+O('x^25); Vec(serlaplace(1/(1 - 9*x)^(19/9))) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = ((9*n+10)(!^9))/10(!^9) = A045756(n+2)/10.
E.g.f.: 1/(1-9*x)^(19/9).
Showing 1-10 of 13 results. Next

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