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 10 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

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

A035274 One fifth of deca-factorial numbers.

Original entry on oeis.org

1, 15, 375, 13125, 590625, 32484375, 2111484375, 158361328125, 13460712890625, 1278767724609375, 134270611083984375, 15441120274658203125, 1930140034332275390625, 260568904634857177734375, 37782491172054290771484375, 5856286131668415069580078125
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

a(n)= (Pochhammer(5/10,n)*10^n)/5.

Links

Crossrefs

Cf. A001147, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

Programs

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

Formula

5*a(n) = (10*n-5)(!^10) = Product_{j=1..n} (10*j-5).
a(n) = 5^n*A001147(n) where A001147(n) = (2*n-1)!!.
E.g.f.: (-1 + (1-10*x)^(-1/2))/5.
a(n) = (Pochhammer(5/10,n)*10^n)/5.
Sum_{n>=1} 1/a(n) = exp(1/10)*sqrt(5*Pi/2)*erf(1/sqrt(10)), where erf is the error function. - 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

A049212 a(n) = -Product_{k=0..n} (10*k - 1); deca-factorial numbers.

Original entry on oeis.org

1, 9, 171, 4959, 193401, 9476649, 559122291, 38579438079, 3047775608241, 271252029133449, 26853950884211451, 2927080646379048159, 348322596919106730921, 44933615002564768288809, 6245772485356502792144451, 930620100318118916029523199, 147968595950580907648694188641
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008543, A035278, A048994, A049209, A049210, A049211, A048176.

Programs

  • Magma
    [Round(10^n*Gamma(n+9/10)/Gamma(9/10)): n in [0..25]]; // G. C. Greubel, Feb 03 2022
  • Mathematica
    CoefficientList[Series[(1-10*x)^(-9/10),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)
  • PARI
    a(n) = {-prod(k=0, n, 10*k-1)} \\ Andrew Howroyd, Jan 02 2020
  • Sage
    [10^n*rising_factorial(9/10, n) for n in (0..25)] # G. C. Greubel, Feb 03 2022

Formula

a(n) = 9*A035278(n) = (10*n-1)(!^10), n >= 1, a(0) = 1.
a(n) = (-1)^n*Sum_{k=0..n} 10^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) = 10^n * Gamma(n+9/10) / Gamma(9/10). - Vaclav Kotesovec, Jan 28 2015
E.g.f.: (1-10*x)^(-9/10). - Vaclav Kotesovec, Jan 28 2015
From Nikolaos Pantelidis, Jan 17 2021: (Start)
G.f.: 1/G(0) where G(k) = 1 - (20*k+9)*x - 10*(k+1)*(10*k+9)*x^2/G(k+1) (continued fraction).
G.f.: 1/(1-9*x-90*x^2/(1-29*x-380*x^2/(1-49*x-870*x^2/(1-69*x-1560*x^2/(1-89*x-2450*x^2/(1-...)))))) (Jacobi continued fraction).
G.f.: 1/Q(0) where Q(k) = 1 - x*(10*k+9)/(1 - x*(10*k+10)/Q(k+1)) (continued fraction).
G.f.: 1/(1-9*x/(1-10*x/(1-19*x/(1-20*x/(1-29*x/(1-30*x/(1-39*x/(1-40*x/(1-49*x/(1-50*x/(1-...))))))))))) (Stieltjes continued fraction).
(End)
G.f.: Hypergeometric2F0([1, 9/10], --; 10*x). - G. C. Greubel, Feb 03 2022
Sum_{n>=0} 1/a(n) = 1 + (e/10)^(1/10)*(Gamma(9/10) - Gamma(9/10, 1/10)). - Amiram Eldar, Dec 22 2022

Extensions

Terms a(14) and beyond from Andrew Howroyd, Jan 02 2020

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

A025755 10th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 45, 2850, 206625, 16116750, 1316201250, 110936962500, 9568313015625, 839885253593750, 74749787569843750, 6727480881285937500, 611079513383472656250, 55937278532794804687500, 5154220664807521289062500, 477624448272163639453125000, 44478776745345238924072265625
Offset: 0

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A035278, A340725.

Programs

  • Mathematica
    CoefficientList[Series[(11 -(1 - 100*x)^(1/10))/10, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 100^(n-1) * Pochhammer[9/10, n-1] / n!; a[0] = 1; Array[a, 26, 0] (* Amiram Eldar, Aug 20 2025 *)

Formula

G.f.: (11-(1-100*x)^(1/10))/10.
a(n) = 10^(n-1)*9*A035278(n-1)/n!, n >= 2, where 9*A035278(n-1) = (10*n-11)(!^10) = Product_{j=2..n} (10*j - 11). - Wolfdieter Lang
Conjecture: n*a(n) + 10*(-10*n+11)*a(n-1) = 0. - R. J. Mathar, Jul 28 2014
a(n) = 100^(n-1)*Pochhammer(9/10, n-1)/n! for n >= 1. Maple confirms this satisfies Mathar's conjecture for n >= 2 (it's not true for n=1). - Robert Israel, Oct 05 2014
a(n) ~ 100^(n-1) / (Gamma(9/10) * n^(11/10)). - Amiram Eldar, Aug 20 2025
Showing 1-10 of 10 results.

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