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 11 results. Next

A035529 A convolution triangle of numbers obtained from A034171.

Original entry on oeis.org

1, 6, 1, 42, 12, 1, 315, 120, 18, 1, 2457, 1134, 234, 24, 1, 19656, 10458, 2673, 384, 30, 1, 160056, 95256, 28539, 5148, 570, 36, 1, 1320462, 861597, 292572, 62532, 8775, 792, 42, 1, 11003850, 7760610, 2920347, 713664, 119565, 13770, 1050, 48, 1
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n,1)= A034171(n-1); a(n,m)=: s2(4; n,m), generalizing s2(2; n,m) := A007318(n-1,m-1) (Pascal), s2(3; n,m) := A035324(n,m).

Examples

			Triangle begins:
      1,
      6,     1;
     42,    12,    1;
    315,   120,   18,   1;
   2457,  1134,  234,  24,  1;
  19656, 10458, 2673, 384, 30, 1;
  ...

Links

Crossrefs

Cf. A034171, A007318, A035324.
Row sums: A049028(n), n >= 1.

Programs

  • Mathematica
    a[n_, m_] /; n - 1 >= m >= 1 := (m*a[n - 1, m - 1])/n + (3*(m + 3*(n - 1))*a[n - 1, m])/n; a[n_, m_] /; n < m = 0; a[n_, 0] = 0; a[n_, n_] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* Jean-François Alcover, Jul 10 2012, from formula *)

Formula

a(n+1, m) = 3*(3*n+m)*a(n, m)/(n+1) + m*a(n, m-1)/(n+1), n >= m >= 1; a(n, m) := 0, n
G.f. for column m: ((-1+(1-9*x)^(-1/3))/3)^m.

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)

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)

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
Showing 1-10 of 11 results. Next

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