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 14 results. Next

A035308 Expansion of 1/(1-100*x)^(1/10), related to deca-factorial numbers A045757.

Original entry on oeis.org

1, 10, 550, 38500, 2983750, 244667500, 20796737500, 1812287125000, 160840482343750, 14475643410937500, 1317283550395312500, 120950580536296875000, 11187928699607460937500, 1041337978963463671875000, 97439482317295529296875000, 9159311337825779753906250000
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A045757, A035024, A004993, A000984, A004987, A256191.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-100*x)^(1/10), {x, 0, 20}], x] (* Amiram Eldar, Aug 18 2025 *)

Formula

a(n) = 10^n*A045757(n)/n!, n >= 1, where A045757(n) = (10*n-9)(!^10) = Product_{j=1..n} (10*j-9).
G.f.: (1-100*x)^(-1/10).
a(n) ~ 10^(2*n) * n^(-9/10) / Gamma(1/10). - Amiram Eldar, Aug 18 2025

A035323 Related to deca-factorial numbers A045757.

Original entry on oeis.org

1, 55, 3850, 298375, 24466750, 2079673750, 181228712500, 16084048234375, 1447564341093750, 131728355039531250, 12095058053629687500, 1118792869960746093750, 104133797896346367187500, 9743948231729552929687500, 915931133782577975390625000, 86441000750730796427490234375
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

Convolution of A035308(n-1) with A025755(n), n >= 1.

Links

Crossrefs

Cf. A045757, A035308, A025755, A256191.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[(-1 + (1 - 100 x)^(-1/10))/10, {x, 0, 13}], x] (* Michael De Vlieger, Oct 13 2019 *)

Formula

a(n) = 10^(n-1)*A045757(n)/n!, where A045757(n) = (10*n-9)(!^10) = Product_{j=1..n} (10*j-9).
G.f.: (-1+(1-100*x)^(-1/10))/10.
D-finite with recurrence: n*a(n) + 10*(-10*n+9)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 10^(2*n-1) * n^(-9/10) / Gamma(1/10). - Amiram Eldar, Aug 18 2025

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

A035279 One tenth of deca-factorial numbers.

Original entry on oeis.org

1, 20, 600, 24000, 1200000, 72000000, 5040000000, 403200000000, 36288000000000, 3628800000000000, 399168000000000000, 47900160000000000000, 6227020800000000000000, 871782912000000000000000, 130767436800000000000000000, 20922789888000000000000000000
Offset: 1

Views

Author

Wolfdieter Lang

Keywords

Comments

E.g.f. is g.f. for A011557(n-1) (powers of ten).

Links

Crossrefs

Cf. A011557, A045757.

Programs

  • GAP
    List([1..20], n-> 10^(n-1)*Factorial(n) ); # G. C. Greubel, Nov 11 2019
  • Magma
    [10^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Nov 11 2019
  • Maple
    seq(10^(n-1)*n!, n=1..20); # G. C. Greubel, Nov 11 2019
  • Mathematica
    Table[10^(n-1)*n!, {n,20}] (* G. C. Greubel, Nov 11 2019 *)
  • PARI
    vector(20, n, 10^(n-1)*n! ) \\ G. C. Greubel, Nov 11 2019
  • Sage
    [10^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Nov 11 2019

Formula

10*a(n) = (10*n)(!^10) = Product_{j=1..n} 10*j = 10^n*n!.
E.g.f.: (-1 + (1-10*x)^(-1))/10.
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 10*(exp(1/10)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 10*(1-exp(-1/10)). (End)

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
Showing 1-10 of 14 results. Next

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