A220605 -id:A220605 - cp's OEIS frontend

A084942 Enneagorials: n-th polygorial for k=9.

1, 1, 9, 216, 9936, 745200, 82717200, 12738448800, 2598643555200, 678245967907200, 220429939569840000, 87290256069656640000, 41375581377017247360000, 23128949989752641274240000, 15056946443328969469530240000, 11292709832496727102147680000000, 9666559616617198399438414080000000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084943, A084944, A085356, A220605.

a := n->n!/2^n*product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[9, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)

a(n)=n!/2^n*prod(i=1,n,7*i-5) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 9) = (A000142(n)/A000079(n))*A084947(n) = (n!/2^n)*Product_{i=0..n-1} (7*i+2) = (n!/2^n)*7^n*Pochhammer(2/7, n) = (n!/2^n)*7^n*Gamma(n+2/7)/Gamma(2/7).
D-finite with recurrence 2*a(n) = n*(7*n-5)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 7^n * n^(2*n + 2/7) * Pi /(Gamma(2/7) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A220086 Decimal expansion of Gamma(1/7).

Original entry on oeis.org

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

A020078 a(n) = floor( Gamma(n+2/7)/Gamma(2/7) ).

Original entry on oeis.org

1, 0, 0, 0, 2, 11, 62, 392, 2862, 23714, 220204, 2264965, 25561751, 314044378, 4172303889, 59604341272, 911094930876, 14837831731419, 256482519928831, 4689966078698628, 90449345803473558, 1834829586299035048

Offset: 0

Simon Plouffe

nonn

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

Cf. A220605.

Cf. A020074, A020075, A020076, A020077, A020079.

[Floor(Gamma(n+2/7)/Gamma(2/7)): 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(2/7,n)), n = 0..25); # G. C. Greubel, Nov 17 2019

Floor[Pochhammer[2/7, Range[0, 25]]] (* G. C. Greubel, Nov 17 2019 *)

vector(26, n, my(x=2/7); gamma(n-1+x)\gamma(x) ) \\ G. C. Greubel, Nov 17 2019

[floor(rising_factorial(2/7, n)) for n in (0..25)] # G. C. Greubel, Nov 17 2019

A220606 Decimal expansion of Gamma(5/7).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 17 2012

See the second comment of A220086.

			1.2759926754934440584853056077898749484545889929110519...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index to sequences related to the Gamma function

Cf. A160389 (see the first comment of A220086).

Mathematica
```
RealDigits[Gamma[5/7], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(gamma(5/7)));
```

Equals Pi*sec(3*Pi/14)/A220605.

A020123 Ceiling of Gamma(n+2/7)/Gamma(2/7).

Original entry on oeis.org

1, 1, 1, 1, 3, 12, 63, 393, 2863, 23715, 220205, 2264966, 25561752, 314044379, 4172303890, 59604341273, 911094930877, 14837831731420, 256482519928832, 4689966078698629, 90449345803473559, 1834829586299035049

Offset: 0

Simon Plouffe

nonn

Robert Israel, Table of n, a(n) for n = 0..450

Cf. A220605.

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

Table[Ceiling[Gamma[n+2/7]/Gamma[2/7]],{n,0,30}] (* Harvey P. Dale, Nov 24 2022 *)

a(n) ~ sqrt(2*Pi)*n^(n-3/14)*exp(-n)/Gamma(2/7). - Robert Israel, Jan 02 2017

A020033 Nearest integer to Gamma(n + 2/7)/Gamma(2/7).

Original entry on oeis.org

1, 0, 0, 1, 3, 12, 62, 393, 2862, 23714, 220205, 2264965, 25561752, 314044379, 4172303889, 59604341272, 911094930877, 14837831731420, 256482519928831, 4689966078698629, 90449345803473559, 1834829586299035049

Offset: 0

Simon Plouffe

Gamma(n + 2/7)/Gamma(2/7) = 1, 2/7, 18/49, 288/343, 6624/2401, 198720/16807, 7352640/117649, 323516160/823543, ... - R. J. Mathar, Sep 04 2016

Cf. A084947, A020078, A020123, A220605.

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

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A084942 Enneagorials: n-th polygorial for k=9.

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A220086 Decimal expansion of Gamma(1/7).

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A020078 a(n) = floor( Gamma(n+2/7)/Gamma(2/7) ).

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A220606 Decimal expansion of Gamma(5/7).

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A020123 Ceiling of Gamma(n+2/7)/Gamma(2/7).

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Maple

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A020033 Nearest integer to Gamma(n + 2/7)/Gamma(2/7).

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Maple