A220607 -id:A220607 - cp's OEIS frontend

A220086 Decimal expansion of Gamma(1/7).

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

Offset: 1

Bruno Berselli, Dec 12 2012

(A220086/A220605)*(A220607/A220606) = A160389, which is the case n=7 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n).
A220086*A220605*A220606*A220607*A220608*A220609 = (2*Pi)^3/sqrt(7), which is the case n=7 of product(Gamma(i/n), i=1..n-1) = sqrt((2*Pi)^(n-1)/n) (see also the second link to Wikipedia).
Continued fraction expansion: 6, 1, 1, 4, 1, 2, 2, 1, 5, 1, 10, 7, 1,...

			6.5480629402478244377140933494289962626211351873841351...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Particular values of the Gamma function: General rational arguments
Wikipedia, Particular values of the Gamma function: Products
Index to sequences related to the Gamma function

SetDefaultRealField(RealField(100)); Gamma(1/7); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[Gamma[1/7], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(gamma(1/7)));
```

default(realprecision, 100); gamma(1/7) \\ G. C. Greubel, Mar 10 2018

Equals Pi * csc(Pi/7) / A220607, where csc is the cosecant function.
(A220086/A220605) * (A220607/A220606) = A160389, which is the case n=7 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n).
A220086*A220605*A220606*A220607*A220608*A220609 = (2*Pi)^3/sqrt(7), which is the case n=7 of product(Gamma(i/n), i=1..n-1) = sqrt((2*Pi)^(n-1)/n) (see also the second link to Wikipedia).
A220086*A220607 = A220605*A220606 + A220608*A220609. - Andrea Pinos, May 21 2025

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

Michel Lagneau, Sep 16 2012

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).

Cf. A004988, A004994, A049382, A216702, A220607.

Cf. A034835, A386271, A386272, A386273, A386274.

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);

Table[49^n * Pochhammer[6/7, n] / n!, {n, 0, 15}] (* Amiram Eldar, Aug 17 2025 *)

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)

A020074 a(n) = floor( Gamma(n+6/7)/Gamma(6/7) ).

Original entry on oeis.org

1, 0, 1, 4, 17, 85, 499, 3422, 26888, 238157, 2347553, 25487720, 302211542, 3885576978, 53842995267, 799953072548, 12684970150410, 213832353964066, 3818434892215473, 72004772253206072, 1429809049027949159

Offset: 0

Simon Plouffe

nonn

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A220607.

Cf. A020075, A020076, A020077, A020078, A020079.

[Floor(Gamma(n+6/7)/Gamma(6/7)): n in [0..25]]; // G. C. Greubel, Nov 17 2019

Digits := 64:f := proc(n,x) trunc(GAMMA(n+x)/GAMMA(x)); end;
seq(floor(pochhammer(6/7,n)), n = 0..25); # G. C. Greubel, Nov 17 2019

Floor[Pochhammer[6/7, Range[0, 25]]] (* G. C. Greubel, Nov 17 2019 *)

vector(26, n, my(x=6/7); gamma(n-1+x)\gamma(x) ) \\ G. C. Greubel, Nov 17 2019

[floor(rising_factorial(6/7, n)) for n in (0..25)] # G. C. Greubel, Nov 17 2019

A025752 7th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 21, 637, 22295, 842751, 33429123, 1370594043, 57564949806, 2462500630590, 106872527367606, 4692675519868518, 208041948047504298, 9297874755046153626, 418404363977076913170, 18939770876029014936162, 861759574859320179595371, 39387481745040692914447251

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. A034833, A220607.

CoefficientList[Series[(8 - (1 - 49*x)^(1/7))/7, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 49^(n-1) * Pochhammer[6/7, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

G.f.: (8-(1-49*x)^(1/7))/7.
a(n) = 7^(n-1)*6*A034833(n-1)/n!, n >= 2, where 6*A034833(n-1)= (7*n-8)(!^7) = Product_{j=2..n} (7*j - 8). - Wolfdieter Lang
a(n) ~ 49^(n-1) / (Gamma(6/7) * n^(8/7)). - Amiram Eldar, Aug 20 2025

A020119 Ceiling of GAMMA(n+6/7)/GAMMA(6/7).

Original entry on oeis.org

1, 1, 2, 5, 18, 86, 500, 3423, 26889, 238158, 2347554, 25487721, 302211543, 3885576979, 53842995268, 799953072549, 12684970150411, 213832353964067, 3818434892215474, 72004772253206073, 1429809049027949160

Offset: 0

Simon Plouffe

nonn

Cf. A220607.

Digits := 64:f := proc(n,x) ceil(GAMMA(n+x)/GAMMA(x)); end;

A020029 Nearest integer to Gamma(n + 6/7)/Gamma(6/7).

Original entry on oeis.org

1, 1, 2, 5, 18, 85, 499, 3422, 26889, 238158, 2347553, 25487720, 302211543, 3885576978, 53842995268, 799953072548, 12684970150411, 213832353964067, 3818434892215474, 72004772253206073, 1429809049027949160

Offset: 0

Simon Plouffe

nonn

Gamma(n + 6/7)/Gamma(6/7) = 1, 6/7, 78/49, 1560/343, 42120/2401, 1432080/16807, 58715280/117649, ... - R. J. Mathar, Sep 04 2016

Cf. A049209, A020074, A020119, A220607.

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

Table[Round[Gamma[n+6/7]/Gamma[6/7]],{n,0,20}] (* Harvey P. Dale, Dec 16 2023 *)

cp's OEIS Frontend

A220086 Decimal expansion of Gamma(1/7).

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

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A020074 a(n) = floor( Gamma(n+6/7)/Gamma(6/7) ).

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A025752 7th-order Patalan numbers (generalization of Catalan numbers).

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A020119 Ceiling of GAMMA(n+6/7)/GAMMA(6/7).

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A020029 Nearest integer to Gamma(n + 6/7)/Gamma(6/7).

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Mathematica