A049393 -id:A049393 - cp's OEIS frontend

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

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

Offset: 0

Joe Keane (jgk(AT)jgk.org)

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

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

A049382 := n -> (-25)^n*binomial(-4/5, n):
seq(A049382(n), n=0..16); # Peter Luschny, Oct 23 2018

CoefficientList[Series[(1-25x)^(-4/5),{x,0,20}],x] (* Harvey P. Dale, Oct 24 2021 *)

a(n) = prod(k=1, n, 25 - 5/k); \\ Michel Marcus, Jun 13 2018

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)

A025750 5th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 10, 150, 2625, 49875, 997500, 20662500, 439078125, 9513359375, 209293906250, 4661546093750, 104884787109375, 2380077861328125, 54401779687500000, 1251240932812500000, 28934946571289062500, 672311993862304687500, 15687279856787109375000, 367412607172119140625000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A034687, A049393, A340722.

CoefficientList[Series[(6-(1-25x)^(1/5))/5,{x,0,20}],x] (* Harvey P. Dale, Dec 06 2012 *)
a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 19 2013, after Vladimir Kruchinin *)
a[n_] := 25^(n-1) * Pochhammer[4/5, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i),i,j,n-k+j-1),j,0,k),k,0,n-1))/(n); /* Vladimir Kruchinin, Dec 10 2011 */

a(n):=if n=0 then 1 else -binomial(1/5,n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */

G.f.: (6-(1-25*x)^(1/5))/5.
a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2, where 4*A034301(n-1) = (5*n-6)(!^5) = Product_{j=2..n} (5*j-6). - Wolfdieter Lang
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n+k-1,n-1) * Sum_{j=0..k} 2^j*binomial(k,j) * Sum_{i=j..n-k+j-1} binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)/n, n > 0, a(0) = 1. - Vladimir Kruchinin, Dec 10 2011
a(n) = ((-5)^(n-1)*Sum_{k=1..n} (5)^(n-k)*stirling1(n,k))/n!, n>0, a(0) = 1. - Vladimir Kruchinin, Mar 19 2013
From Karol A. Penson, Feb 05 2025: (Start)
a(n) without the initial 1 (i.e., a(n) for n >= 1) is given by
a(n+1) = 5^(2*n)*gamma(n + 4/5)/(gamma(4/5)*(n + 1)!), n >= 0.
a(n+1) = Integral_{x=0..25} x^n*W(x) dx, n >= 0,
  where W(x) = sin(Pi/5)*5^(2/5)*(1 - x/25)^(1/5)/(5*Pi*x^(1/5)). W(x) is a positive  function on x = (0, 25), is singular at x = 0 with the singularity (x)^(-1/5), and it goes to zero at x = 25. (End)
a(n) ~ 25^(n-1) / (Gamma(4/5) * n^(6/5)). - Amiram Eldar, Aug 20 2025

A301271 Expansion of (1-16*x)^(1/8).

Original entry on oeis.org

1, -2, -14, -140, -1610, -19964, -259532, -3485144, -47920730, -670890220, -9526641124, -136837208872, -1984139528644, -28998962341720, -426699017313880, -6315145456245424, -93937788661650682, -1403541077650545484, -21053116164758182260, -316904801216886322440

Offset: 0

Seiichi Manyama, Jun 15 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..833

Cf. A003557, A007947.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), this sequence (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

PARI
```
N=20; x='x+O('x^N); Vec((1-16*x)^(1/8))
```

a(n) = 2^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = -sqrt(2-sqrt(2)) * Gamma(1/8) * Gamma(n-1/8) * 16^(n-1) / (Pi*Gamma(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) ~ -2^(4*n-3) / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +2*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020
a(n) = -2*A097184(n-1). - R. J. Mathar, Jan 20 2020

A303007 Expansion of (1-240*x)^(1/8).

Original entry on oeis.org

1, -30, -3150, -472500, -81506250, -15160162500, -2956231687500, -595469525625000, -122815589660156250, -25791273828632812500, -5493541325498789062500, -1183608449221102734375000, -257434837705589844726562500, -56437637496994696728515625000

Offset: 0

Seiichi Manyama, Jun 15 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A003557, A007947, A049210, A108091.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), A004996 (b=36), this sequence (b=240), A303055 (b=504), A305886 (b=1728).

CoefficientList[Series[Surd[1-240x,8],{x,0,20}],x] (* Harvey P. Dale, Aug 29 2024 *)

N=20; x='x+O('x^N); Vec((1-240*x)^(1/8))

a(n) = 30^n/n! * Product_{k=0..n-1} (8*k - 1) for n > 0.
a(n) = 15^n * A301271(n).
a(n) ~ -2^(4*n - 3) * 15^n / (Gamma(7/8) * n^(9/8)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +30*(-8*n+9)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A305991 Expansion of (1-27*x)^(1/9).

Original entry on oeis.org

1, -3, -36, -612, -11934, -250614, -5513508, -125235396, -2911722957, -68910776649, -1653858639576, -40143659706072, -983519662798764, -24285370135261788, -603664914790793016, -15091622869769825400, -379177024602966863175, -9568643738510163782475

Offset: 0

Seiichi Manyama, Jun 16 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..700

Cf. A003557, A007947.

(1-b*x)^(1/A003557(b)): A002420 (b=4), A004984 (b=8), A004990 (b=9), (-1)^n * A108735 (b=12), A301271 (b=16), (-1)^n * A108733 (b=18), A049393 (b=25), this sequence (b=27), A004996 (b=36), A303007 (b=240), A303055 (b=504), A305886 (b=1728).

PARI
```
N=20; x='x+O('x^N); Vec((1-27*x)^(1/9))
```

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k - 1) for n > 0.
a(n) ~ 27^n / (Gamma(-1/9) * n^(10/9)). - Vaclav Kotesovec, Jun 16 2018
D-finite with recurrence: n*a(n) +3*(-9*n+10)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

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

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A025750 5th-order Patalan numbers (generalization of Catalan numbers).

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Maxima

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A301271 Expansion of (1-16*x)^(1/8).

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A303007 Expansion of (1-240*x)^(1/8).

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A305991 Expansion of (1-27*x)^(1/9).

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