A203146 -id:A203146 - cp's OEIS frontend

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

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

Offset: 0

Michel Lagneau, Sep 16 2012

nonn

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).

Harvey P. Dale, Table of n, a(n) for n = 0..553

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

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);

Table[Product[64-8/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Sep 23 2017 *)

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)

A025753 8th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 28, 1120, 51520, 2555392, 132880384, 7137574912, 392566620160, 21983730728960, 1248675905404928, 71742106565083136, 4161042180774821888, 243260927491451125760, 14317643160925409116160, 847604475126784219676672, 50432466270043661070761984, 3014081513550844685170245632

Offset: 0

Olivier Gérard

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. A034975, A203146.

CoefficientList[Series[(9 - (1 - 64*x)^(1/8))/8, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 64^(n-1) * Pochhammer[7/8, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

G.f.: (9-(1-64*x)^(1/9))/8.
a(n) = 8^(n-1)*7*A034975(n-1)/n!, n >= 2, where 7*A034975(n-1)= (8*n-9)!^8 = Product_{j=2..n} (8*j - 9). - Wolfdieter Lang
a(n) ~ 64^(n-1) / (Gamma(7/8) * n^(9/8)). - Amiram Eldar, Aug 20 2025

A242011 Decimal expansion of sum_{k>=0} (-1)^k*(log(4k+1)/(4k+1)+log(4k+3)/(4k+3)).

Original entry on oeis.org

0, 2, 3, 0, 0, 4, 5, 8, 7, 8, 6, 2, 7, 3, 6, 0, 1, 0, 3, 1, 7, 9, 9, 2, 6, 0, 2, 1, 4, 5, 1, 4, 6, 9, 6, 2, 3, 1, 8, 6, 6, 7, 6, 4, 1, 4, 7, 5, 0, 8, 8, 3, 2, 9, 0, 9, 6, 3, 8, 0, 0, 6, 2, 0, 6, 5, 8, 1, 4, 5, 4, 7, 6, 3, 5, 4, 5, 5, 9, 4, 1, 4, 0, 3, 1, 5, 6, 6, 2, 3, 6, 1, 5, 5, 8, 9, 1, 9, 6, 7

Offset: 0

Jean-François Alcover, Aug 11 2014

			0.02300458786273601031799260214514696231866764147508832909638...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 8.

Cf. A203142, A203143, A203144, A203146.

s = (Pi/(2*Sqrt[2]))*(Log[Gamma[1/8]*Gamma[3/8]/(Gamma[5/8]*Gamma[7/8])] - (EulerGamma + Log[2*Pi])); Join[{0}, RealDigits[s, 10, 99] // First]

(Pi/(2*sqrt(3)))*(log(Gamma(1/8)/Gamma(3/8)/(Gamma(5/8)/Gamma(7/8))) - (gamma + log(2*Pi))), where gamma is Euler's constant and Gamma(x) is the Euler Gamma function.

A203127 Decimal expansion of (3/8)! = Gamma(11/8).

Original entry on oeis.org

8, 8, 8, 9, 1, 3, 5, 6, 9, 1, 5, 6, 2, 2, 5, 3, 4, 0, 7, 4, 2, 4, 2, 7, 5, 6, 4, 0, 6, 6, 2, 4, 4, 6, 9, 1, 2, 0, 7, 7, 7, 5, 3, 0, 1, 2, 5, 9, 5, 9, 6, 8, 7, 0, 4, 1, 5, 6, 7, 2, 6, 0, 0, 5, 0, 2, 4, 3, 5, 5, 7, 4, 2, 5, 9, 2, 5, 0, 6, 7, 1, 9, 2, 4, 4, 9, 2, 8, 7, 5, 6, 3, 9, 1, 3, 0, 5, 5, 2

Offset: 0

N. J. A. Sloane, Dec 29 2011

			0.88891356915622534074242756406624469120777530125959687041567...

Index to sequences related to the Gamma function.

Cf. A011027, A203130, A203143, A203146, A231863.

RealDigits[Gamma[11/8], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

gamma(11/8) \\ Charles R Greathouse IV, Mar 25 2014

Equals 3*A203143/8. - R. J. Mathar, Jan 15 2021
A203146 * this * A231863 * A011027 = A203130. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^(8/3)) dx. - Ilya Gutkovskiy, Apr 10 2024

A203132 Decimal expansion of (7/8)! = Gamma(15/8).

Original entry on oeis.org

9, 5, 3, 4, 4, 5, 8, 1, 2, 7, 4, 5, 0, 3, 4, 8, 3, 2, 3, 4, 5, 8, 2, 9, 6, 6, 0, 7, 1, 5, 0, 3, 1, 3, 4, 9, 7, 5, 4, 3, 8, 6, 3, 0, 9, 2, 9, 6, 6, 1, 9, 4, 8, 0, 4, 4, 9, 4, 9, 1, 0, 3, 7, 8, 4, 6, 9, 2, 3, 1, 1, 5, 2, 8, 2, 6, 4, 4, 5, 6, 5, 4, 3, 9, 6, 4, 0, 9, 6, 5, 4, 8, 1, 6, 6, 1, 2, 1, 4

Offset: 0

N. J. A. Sloane, Dec 29 2011

			0.95344581274503483234582966071503134975438630929661948044949...

Index to sequences related to the Gamma function

RealDigits[(7/8)!,10,120][[1]] (* Harvey P. Dale, Jun 23 2016 *)

gamma(15/8) \\ Charles R Greathouse IV, Mar 25 2014

Equals 7*A203146/8. - R. J. Mathar, Jan 15 2021
A203127 * this * A231863*2^(9/4) = (7/4)* A203130. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^(8/7)) dx. - Ilya Gutkovskiy, Apr 10 2024

cp's OEIS Frontend

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

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A025753 8th-order Patalan numbers (generalization of Catalan numbers).

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A242011 Decimal expansion of sum_{k>=0} (-1)^k*(log(4k+1)/(4k+1)+log(4k+3)/(4k+3)).

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A203127 Decimal expansion of (3/8)! = Gamma(11/8).

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A203132 Decimal expansion of (7/8)! = Gamma(15/8).

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