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 1-8 of 8 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

A035278 One ninth of deca-factorial numbers.

Original entry on oeis.org

1, 19, 551, 21489, 1052961, 62124699, 4286604231, 338641734249, 30139114348161, 2983772320467939, 325231182931005351, 38702510768789636769, 4992623889173863143201, 693974720595166976904939, 103402233368679879558835911, 16440955105620100849854909849
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045757, A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-1)/9 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-1: j in [1..n]])/9: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-1, j=1..n)/9, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[9/10, n]/9, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-1)/9 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-1) for j in (1..n))/9 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

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

A035273 One quarter of deca-factorial numbers.

Original entry on oeis.org

1, 14, 336, 11424, 502656, 27143424, 1737179136, 128551256064, 10798305509376, 1015040717881344, 105564234659659776, 12034322751201214464, 1492256021148950593536, 199962306833959379533824, 28794572184090150652870656, 4434364116349883200542081024
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034323, A035272, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-6)/4 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-6: j in [1..n]])/4: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-6, j=1..n)/4, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[4/10, n]/4, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-6)/4 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-6) for j in (1..n))/4 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

4*a(n) = (10*n-6)(!^10) = Product_{j=1..n} (10*j-6).
a(n) = 2^(n+1)*A034323(n) where 2*A034323(n)= (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).
E.g.f.: (-1 + (1-10*x)^(-2/5))/4.
a(n) = (Pochhammer(4/10,n)*10^n)/4.
Sum_{n>=1} 1/a(n) = 4*(e/10^6)^(1/10)*(Gamma(2/5) - Gamma(2/5, 1/10)). - Amiram Eldar, Dec 22 2022

A035277 One eighth of deca-factorial numbers.

Original entry on oeis.org

1, 18, 504, 19152, 919296, 53319168, 3625703424, 282804867072, 24886828302336, 2438909173628928, 263402190751924224, 31081458508727058432, 3978426689117063479296, 549022883098154760142848, 81255386698526904501141504, 12838351098367250911180357632
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034301, A045757, A035265, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-2)/8 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-2: j in [1..n]])/8: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-2, j=1..n)/8, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[4/5, n]/8, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-2)/8 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-2) for j in (1..n))/8 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

a(n) = (Pochhammer(8/10,n)*10^n)/8.
8*a(n) = (10*n-2)(!^10) = Product_{j=1..n} (10*j-2).
a(n) = 2^(n+2)*A034301(n) where 4*A034301(n) = (5*n-1)(!^5).
E.g.f.: (-1 + (1-10*x)^(-4/5))/8.
Sum_{n>=1} 1/a(n) = 8*(e/10^2)^(1/10)*(Gamma(4/5) - Gamma(4/5, 1/10)). - Amiram Eldar, Dec 22 2022

A035272 One third of deca-factorial numbers.

Original entry on oeis.org

1, 13, 299, 9867, 424281, 22486893, 1416674259, 103417220907, 8583629335281, 798277528181133, 82222585402656699, 9291152150500206987, 1142811714511525459401, 151993958030032886100333, 21735135998294702712347619, 3325475807739089514989185707, 542052556661471590943237270241
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-7)/3 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-7: j in [1..n]])/3: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-7, j=1..n)/3, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[3/10, n]/3, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-7)/3 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-7) for j in (1..n))/3 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

3*a(n) = (10*n-7)(!^10) = Product_{j=1..n} (10*j-7).
E.g.f.: (-1 + (1-10*x)^(-3/10))/3.
Sum_{n>=1} 1/a(n) = 3*(e/10^7)^(1/10)*(Gamma(3/10) - Gamma(3/10, 1/10)). - Amiram Eldar, Dec 22 2022

A035275 One sixth of deca-factorial numbers.

Original entry on oeis.org

1, 16, 416, 14976, 688896, 38578176, 2546159616, 193508130816, 16641699250176, 1597603128016896, 169345931569790976, 19644128062095753216, 2475160135824064905216, 336621778472072827109376, 49146779656922632757968896, 7666897626479930710243147776
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A034300, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-4)/6 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-4: j in [1..n]])/6: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-4, j=1..n)/6, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[6/10, n]/6, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-4)/6 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-4) for j in (1..n))/6 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

6*a(n) = (10*n-4)(!^10) = Product_{j=1..n} (10*j-4).
a(n) = 2^n*3*A034300(n) where 3*A034300(n) = (5*n-2)(!^5).
E.g.f.: (-1 + (1-10*x)^(-3/5))/6.
a(n) = (Pochhammer(6/10,n) * 10^n)/6.
Sum_{n>=1} 1/a(n) = 6*(e/10^4)^(1/10)*(Gamma(3/5) - Gamma(3/5, 1/10)). - Amiram Eldar, Dec 22 2022

A035276 One seventh of deca-factorial numbers.

Original entry on oeis.org

1, 17, 459, 16983, 798201, 45497457, 3048329619, 234721380663, 20420760117681, 1980813731415057, 211947069261411099, 24797807103585098583, 3149321502155307520041, 431457045795277130245617, 63424185731905738146105699, 9957597159909200888938594743
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-3)/7 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-3: j in [1..n]])/7: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-3, j=1..n)/7, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[7/10, n]/7, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-3)/7 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-3) for j in (1..n))/7 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

7*a(n) = (10*n-3)(!^10) = Product_{j=1..n} (10*j-3).
E.g.f.: (-1 + (1-10*x)^(-7/10))/7.
a(n) = (Pochhammer(7/10,n)*10^n)/7.
Sum_{n>=1} 1/a(n) = 7*(e/10^3)^(1/10)*(Gamma(7/10) - Gamma(7/10, 1/10)). - Amiram Eldar, Dec 22 2022

A035265 One half of deca-factorial numbers.

Original entry on oeis.org

1, 12, 264, 8448, 354816, 18450432, 1143926784, 82362728448, 6753743732736, 621344423411712, 63377131187994624, 7098238693055397888, 865985120552758542336, 114310035912964127588352, 16232025099640906117545984, 2467267815145417729866989568, 399697386053557672238452310016
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008548, A045757, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279.

Programs

  • GAP
    List([1..20], n-> Product([1..n], j-> 10*j-8)/2 ); # G. C. Greubel, Nov 11 2019
  • Magma
    [(&*[10*j-8: j in [1..n]])/2: n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq( mul(10*j-8, j=1..n)/2, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^n*Pochhammer[2/10, n]/2, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, prod(j=1,n, 10*j-8)/2 ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [product( (10*j-8) for j in (1..n))/2 for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

2*a(n) = (10*n-8)(!^10) = Product_{j=1..n} (10*j-8).
a(n) = 2^n*A008548(n) where A008548(n) = (5*n-4)(!^5) = Product_{j=1..n} (5*j-4).
E.g.f.: (-1 + (1-10*x)^(-1/5))/2.
a(n) = (Pochhammer(2/10,n)*10^n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/10^8)^(1/10)*(Gamma(1/5) - Gamma(1/5, 1/10)). - Amiram Eldar, Dec 22 2022
Showing 1-8 of 8 results.

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