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

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

A216702 a(n) = Product_{k=1..n} (16 - 4/k).

Original entry on oeis.org

1, 12, 168, 2464, 36960, 561792, 8614144, 132903936, 2060011008, 32044615680, 499896004608, 7816555708416, 122459372765184, 1921670157238272, 30197673899458560, 475110069351481344, 7482983592285831168, 117967035454858985472, 1861257670509997326336
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Links

Crossrefs

Cf. A004988, A049382, A004994, A034385, A098430.

Programs

  • Maple
    seq(product(16-4/k, k=1.. n), n=0..20);
seq((4^n/n!)*product(4*k+3, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[16-4/k,{k,n}],{n,0,20}] (* or *) CoefficientList[ Series[ 1/(1-16*x)^(3/4),{x,0,20}],x] (* Harvey P. Dale, Sep 19 2012 *)

Formula

G.f.: 1/(1-16*x)^(3/4). - Harvey P. Dale, Sep 19 2012
From Peter Bala, Sep 24 2023: (Start)
a(n) = 16^n * binomial(n - 1/4, n).
P-recursive: a(n) = 4*(4*n - 1)/n * a(n-1) with a(0) = 1. (End)
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n * binomial(-3/4, n).
a(n) ~ 1/Gamma(3/4) * 16^n/n^(1/4).
E.g.f.: hypergeom([3/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-3/4, k)* binomial(-3/4, 2*n - k).
(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = (16^n)/(2*n)! * Product_{k = 1..n} (4*k^2 - 1) = (16^n)/(2*n)! * A079484(n). (End)

A216703 a(n) = Product_{k=1..n} (49 - 7/k).

Original entry on oeis.org

1, 42, 1911, 89180, 4213755, 200574738, 9594158301, 460519598448, 22162505675310, 1068725273676060, 51619430718553698, 2496503376570051576, 120872371815599997138, 5857661095679076784380, 284096563140435224042430, 13788153197749122873525936
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A004994, A049382, A216702, A220607.
Cf. A034835, A386271, A386272, A386273, A386274.

Programs

  • Maple
    seq(product(49-7/k, k=1.. n), n=0..20);
seq((7^n/n!)*product(7*k+6, k=0.. n-1), n=0..20);
  • Mathematica
    Table[49^n * Pochhammer[6/7, n] / n!, {n, 0, 15}] (* Amiram Eldar, Aug 17 2025 *)

Formula

From Seiichi Manyama, Jul 17 2025: (Start)
G.f.: 1/(1 - 49*x)^(6/7).
a(n) = (-49)^n * binomial(-6/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+6). (End)
From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 49^n * Gamma(n+6/7) / (Gamma(6/7) * Gamma(n+1)).
a(n) ~ c * 49^n / n^(1/7), where c = 1/Gamma(6/7) = 1/A220607 = 0.904349... . (End)

A049381 Expansion of (1-25*x)^(-3/5).

Original entry on oeis.org

1, 15, 300, 6500, 146250, 3363750, 78487500, 1850062500, 43938984375, 1049653515625, 25191684375000, 606890578125000, 14666522304687500, 355381117382812500, 8630684279296875000, 210013317462890625000
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Examples

			(1-x)^(-3/5) = 1 + 3/5*x + 12/25*x^2 + 52/125*x^3 + ...

Links

Crossrefs

Cf. A034688, A049380, A049382, A049392, A049396.

Programs

Formula

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

A216704 a(n) = Product_{k=1..n} (64 - 8/k).

Original entry on oeis.org

1, 56, 3360, 206080, 12776960, 797282304, 49963024384, 3140532961280, 197853576560640, 12486759054049280, 789163172215914496, 49932506169297862656, 3162392057388864634880, 200447004252955727626240, 12714067126901763295150080, 806919460320698577132191744
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Links

Crossrefs

Cf. A004988, A049382, A004994, A203146, A216702, A216703.

Programs

  • Maple
    seq(product(64-8/k, k=1.. n), n=0..20);
seq((8^n/n!)*product(8*k+7, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[64-8/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Sep 23 2017 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 64^n * Gamma(n+7/8) / (Gamma(7/8) * Gamma(n+1)).
a(n) ~ c * 64^n / n^(1/8), where c = 1/Gamma(7/8) = 1/A203146 = 0.917723... . (End)

A049393 Expansion of (1-25*x)^(1/5).

Original entry on oeis.org

1, -5, -50, -750, -13125, -249375, -4987500, -103312500, -2195390625, -47566796875, -1046469531250, -23307730468750, -524423935546875, -11900389306640625, -272008898437500000, -6256204664062500000, -144674732856445312500, -3361559969311523437500
Offset: 0

Views

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

Links

Crossrefs

Cf. A049382.

Programs

  • Maple
    A049393 := proc(n)
    coeftayl((1-25*x)^(1/5), x=0, n);
end proc:
seq(A049393(n), n=0..20); # Wesley Ivan Hurt, Jul 13 2014
  • Mathematica
    CoefficientList[Series[(1 - 25 x)^(1/5), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jul 13 2014 *)
  • PARI
    Vec(taylor((1-25*x)^(1/5), x, 40)) \\ Michel Marcus, Jul 16 2014

Formula

G.f.: (1-25*x)^(1/5).
a(n) = 5^n/n! * product[ k=0..n-1 ] (5*k-1).
a(n) ~ -1/5*Gamma(4/5)^-1*n^(-6/5)*5^(2*n)*{1 + 3/25*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

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)

A216705 a(n) = Product_{k=1..n} (81 - 9/k).

Original entry on oeis.org

1, 72, 5508, 429624, 33832890, 2679564888, 213025408596, 16981168285224, 1356370816782267, 108509665342581360, 8691624193940766936, 696910230823250585232, 55927046023565859464868, 4491372003738673637024784, 360913821729000560118063000
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Links

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704.

Programs

  • Maple
    seq(product(81-9/k, k=1.. n), n=0..20);
seq((9^n/n!)*product(9*k+8, k=0.. n-1), n=0..20);
  • Mathematica
    Table[Product[81-9/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 20 2021 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 81^n * Gamma(n+8/9) / (Gamma(8/9) * Gamma(n+1)).
a(n) ~ c * 81^n / n^(1/9), where c = 1/Gamma(8/9) = 0.927851... . (End)

A216706 a(n) = Product_{k=1..n} (100 - 10/k).

Original entry on oeis.org

1, 90, 8550, 826500, 80583750, 7897207500, 776558737500, 76546504125000, 7558967282343750, 747497875698437500, 74002289694145312500, 7332954160601671875000, 727184620926332460937500, 72159089307305298046875000, 7164366724082454591796875000
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A340725.

Programs

  • Maple
    seq(product(100-10/k, k=1.. n), n=0..20);
seq((10^n/n!)*product(10*k+9, k=0.. n-1), n=0..20);

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 100^n * Gamma(n+9/10) / (Gamma(9/10) * Gamma(n+1)).
a(n) ~ c * 100^n / n^(1/10), where c = 1/Gamma(9/10) = 1/A340725 = 0.935778... . (End)

A216786 a(n) = Product_{k=1..n} (121 - 11/k).

Original entry on oeis.org

1, 110, 12705, 1490720, 176277640, 20941783632, 2495562549480, 298041470195040, 35653210872081660, 4270462368900447720, 512028438031163681628, 61443412563739641795360, 7378329792029068652259480, 886534702703800402679177520, 106574136046464005550646840440
Offset: 0

Views

Author

Michel Lagneau, Sep 16 2012

Keywords

Comments

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Crossrefs

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706.

Programs

  • Maple
    seq(product(121-11/k, k=1.. n), n=0..20);
seq((11^n/n!)*product(11*k+10, k=0.. n-1), n=0..20);
A216786 := proc(n)
    binomial(-10/11,n)*(-121)^n ;
end proc: # R. J. Mathar, Sep 17 2012
  • Mathematica
    Join[{1},FoldList[Times,121-11/Range[20]]] (* Harvey P. Dale, Mar 15 2016 *)

Formula

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 121^n * Gamma(n+10/11) / (Gamma(10/11) * Gamma(n+1)).
a(n) ~ c * 121^n / n^(1/11), where c = 1/Gamma(10/11) = 0.942148... . (End)
Showing 1-10 of 14 results. Next

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