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 1-6 of 6 results.

A034688 Expansion of (1-25*x)^(-1/5), related to quintic factorial numbers A008548.

Original entry on oeis.org

1, 5, 75, 1375, 27500, 577500, 12512500, 277062500, 6233906250, 141994531250, 3265874218750, 75708902343750, 1766541054687500, 41445770898437500, 976936028320312500, 23120819336914062500, 549119459251708984375
Offset: 0

Views

Author

Wolfdieter Lang

Keywords

Links

Crossrefs

Cf. A008548, A034385, A034687, A049380, A049381, A049382.

Programs

  • GAP
    List([0..20], n-> 5^n*Product([0..n-1], k-> 5*k+1)/Factorial(n)); # G. C. Greubel, Aug 17 2019
  • Magma
    [1] cat [5^n*(&*[5*k+1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 17 2019
  • Maple
    A034688 := n -> (-25)^n*binomial(-1/5, n):
seq(A034688(n), n=0..16); # Peter Luschny, Oct 23 2018
  • Mathematica
    Table[(-25)^n*Binomial[-1/5,n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *)
CoefficientList[Series[1/Surd[1-25x,5],{x,0,20}],x] (* Harvey P. Dale, Sep 11 2022 *)
  • PARI
    vector(20, n, n--; 5^n*prod(k=0, n-1, 5*k+1)/n!) \\ G. C. Greubel, Aug 17 2019
  • Sage
    [5^n*product(5*k+1 for k in (0..n-1))/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 17 2019

Formula

a(n) = (5^n/n!)*A008548(n), n >= 1, a(0) := 1, where A008548(n)=(5*n-4)(!^5) := Product_{j=1..n} (5*j-4).
G.f.: (1-25*x)^(-1/5).
a(n) ~ Gamma(1/5)^-1*n^(-4/5)*5^(2*n)*{1 - 2/25*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = (-25)^n*binomial(-1/5, n). - Peter Luschny, Oct 23 2018
E.g.f.: L_{-1/5}(25*x), where L_{k}(x) is the Laguerre polynomial. - Stefano Spezia, Aug 17 2019
D-finite with recurrence: n*a(n) +5*(-5*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

A049382 Expansion of (1-25*x)^(-4/5).

Original entry on oeis.org

1, 20, 450, 10500, 249375, 5985000, 144637500, 3512625000, 85620234375, 2092939062500, 51277007031250, 1258617445312500, 30941012197265625, 761624915625000000, 18768613992187500000, 462959145140625000000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(1-x)^(-4/5) = 1 + 4/5*x + 18/25*x^2 + 84/125*x^3 + ...

Crossrefs

Cf. A008546, A034688, A049393, A049397, A049381, A049382.

Programs

Formula

G.f.: (1-25*x)^(-4/5).
a(n) = (5^n/n!) * Product_{k=0..n-1} (5*k + 4).
a(n) ~ Gamma(4/5)^-1*n^(-1/5)*5^(2*n)*{1 - 2/25*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Product_{k=1..n} (25 - 5/k). - Michel Lagneau, Sep 16 2012
a(n) = (-25)^n*binomial(-4/5, n). - Peter Luschny, Oct 23 2018
From Peter Bala, Sep 24 2023: (Start)
a(n) = 25^n * binomial(n - 1/5, n).
P-recursive: a(n) = 5*(5*n - 1)/n * a(n-1) with a(0) = 1. (End)

A049380 Expansion of (1-25*x)^(-2/5).

Original entry on oeis.org

1, 10, 175, 3500, 74375, 1636250, 36815625, 841500000, 19459687500, 454059375000, 10670395312500, 252209343750000, 5989971914062500, 142837791796875000, 3417904303710937500, 82029703289062500000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(1-x)^(-2/5) = 1 + 2/5*x + 7/25*x^2 + 28/125*x^3 + ...

Links

Crossrefs

Cf. A034688, A049381, A049382, A049391, A049395, A049382.

Programs

  • Maple
    f:= gfun:-rectoproc({a(n+1) = (10+25*n)*a(n)/(n+1),a(0)=1},a(n),remember):
map(f, [$0..50]); # Robert Israel, Sep 04 2018
  • Mathematica
    CoefficientList[Series[1/Surd[(1-25x)^2,5],{x,0,20}],x] (* Harvey P. Dale, Jan 15 2024 *)
  • PARI
    x='x+O('x^99); Vec((1-25*x)^(-2/5)) \\ Altug Alkan, Sep 04 2018

Formula

G.f.: (1-25*x)^(-2/5).
a(n) = (5^n/n!) * Product_{k=0..n-1} (5*k + 2).
a(n) ~ Gamma(2/5)^(-1)*n^(-3/5)*5^(2*n)*{1 - 3/25*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n+1) = (10 + 25*n)*a(n)/(n+1). - Robert Israel, Sep 04 2018
a(n) = (-25)^n*binomial(-2/5,n). - Peter Luschny, Oct 23 2018
From Peter Bala, Sep 24 2023: (Start)
a(n) = 25^n * binomial(n - 3/5, n).
P-recursive: a(n) = 5*(5*n - 3)/n * a(n-1) with a(0) = 1. (End)

A386272 Expansion of 1/(1 - 49*x)^(3/7).

Original entry on oeis.org

1, 21, 735, 29155, 1224510, 53143734, 2356038874, 106021749330, 4823989594515, 221367522503855, 10227179539678101, 475098976797773601, 22171285583896101380, 1038639455430209672340, 48816054405219854599980, 2300863364299362480145724, 108715793963144877186885459
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A034835, A216703, A386271, A386273, A386274.
Cf. A049381, A216702.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(3/7))

Formula

a(n) = (-49)^n * binomial(-3/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+3).
a(n) = 7^n * Product_{k=1..n} (7 - 4/k).
D-finite with recurrence n*a(n) +7*(-7*n+4)*a(n-1)=0. - R. J. Mathar, Jul 20 2025

A049392 Expansion of (1-25*x)^(2/5).

Original entry on oeis.org

1, -10, -75, -1000, -16250, -292500, -5606250, -112125000, -2312578125, -48821093750, -1049653515625, -22901531250000, -505742148437500, -11281940234375000, -253843655273437500, -5753789519531250000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(1-x)^(2/5) = 1 - 2/5*x - 3/25*x^2 - 8/125*x^3 - ...

Links

Crossrefs

Cf. A049381.

Programs

  • PARI
    first(n) = x='x+O('x^n); Vec((1-25*x)^(2/5)) \\ Iain Fox, Aug 10 2018

Formula

G.f.: (1-25*x)^(2/5).
a(n) = (5^n/n!) * Product_{k=0..n-1} (5*k-2).
a(n) ~ -2/5*Gamma(3/5)^-1*n^(-7/5)*5^(2*n)*{1 + 7/25*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

Extensions

Definition corrected by Georg Fischer, Jan 16 2020

A049396 Expansion of (1-25*x)^(-8/5).

Original entry on oeis.org

1, 40, 1300, 39000, 1121250, 31395000, 863362500, 23434125000, 629792109375, 16794456250000, 445053090625000, 11733217843750000, 307996968398437500, 8055305327343750000, 210013317462890625000, 5460346254035156250000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(1-25*x)^(-8/5) = 1+40*x+1300*x^2+39000*x^3+1121250*x^4+31395000*x^5+...

Crossrefs

Cf. A049381.

Programs

  • Mathematica
    CoefficientList[Series[(1-25x)^-(8/5),{x,0,20}],x] (* Harvey P. Dale, May 06 2016 *)

Formula

G.f.: (1-25*x)^(-8/5).
a(n) = (5^n/n!) * product[ k=0..n-1 ] (5*k+8).
a(n) ~ (5/3)*Gamma(3/5)^-1*n^(3/5)*5^(2*n)*{1 + 12/25*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

Extensions

Typo in definition corrected by N. J. A. Sloane, May 06 2016
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.