A246745 -id:A246745 - cp's OEIS frontend

A084940 Heptagorials: n-th polygorial for k=7.

1, 1, 7, 126, 4284, 235620, 19085220, 2137544640, 316356606720, 59791398670080, 14050978687468800, 4018579904616076800, 1374354327378698265600, 553864793933615401036800, 259762588354865623086259200, 140271797711627436466579968000, 86407427390362500863413260288000

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, A084941, A084942, A084943, A084944, A085356, A246745.

a := n->n!/2^n*mul(5*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[7, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)
Join[{1},FoldList[Times,PolygonalNumber[7,Range[20]]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 29 2019 *)

a(n)=n!/2^n*prod(i=1,n,5*i-3) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 7) = (A000142(n)/A000079(n))*A047055(n) = (n!/2^n)*Product_{i=0..n-1}(5*i+2) = (n!/2^n)*5^n*Pochhammer(2/5, n) = (n!/2^n)*5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi.
D-finite with recurrence 2*a(n) = n*(5*n-3)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 5^n * n^(2*n + 2/5) * Pi /(Gamma(2/5) * 2^(n-1) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A256191 Decimal expansion of Gamma(1/10).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 19 2015

			9.513507698668731836292487177265402192550578626088377...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index to sequences related to the Gamma function

Cf. A175380, A246745, A340721, A340722.

Cf. A340723, A340724, A340725.

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

Maple
```
evalf(GAMMA(1/10),100);
```
Mathematica
```
RealDigits[Gamma[1/10],10,100][[1]]
```
PARI
```
gamma(1/10)
```

From Vaclav Kotesovec, Apr 10 2024: (Start)
Equals 5^(1/4) * sqrt(1 + sqrt(5)) * Gamma(1/5) * Gamma(2/5) / (2^(7/10) * sqrt(Pi)).
Equals 2^(4/5) * sqrt(Pi) * Gamma(1/5) / Gamma(3/5). (End)

A340721 Decimal expansion of Gamma(3/5).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.489192248812817102..

DLMF, Gamma function properties, DLMF section 5.5.
Eric Weisstein's World of Mathematics, Gamma Function.
Wikipedia, Particular values of the Gamma function.
Index to sequences related to gamma function.

Cf. A019881, A175380, A246745, A256191.

Maple
```
evalf(GAMMA(3/5),120) ;
```

RealDigits[Gamma[3/5], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A246745 = Pi/A019881. [DLMF (5.5.3)]

this * A256191 *2^(7/10)/sqrt(2*Pi) = 2*A175380 [DLMF (5.5.5)]

A340725 Decimal expansion of Gamma(9/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.06862870211931935489..

DLMF, Gamma function properties, DLMF section 5.5.
Index to sequences related to gamma function.

Cf. A019827, A246745, A256191, A340722.

Maple
```
evalf(GAMMA(9/10),120) ;
```

RealDigits[Gamma[9/10], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

this * A256191 = Pi/A019827 . [DLMF (5.5.3)]

A246745 * this *2^(3/10) /sqrt(2*Pi) = A340722 . [DLMF (5.5.5)]

A340724 Decimal expansion of Gamma(7/10).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 17 2021

			1.29805533264755778568...

DLMF, Gamma function properties, DLMF section 5.5.
Index to sequences related to gamma function

Cf. A246745, A175380, A246745.

Maple
```
evalf(GAMMA(7/10),120) ;
```

RealDigits[Gamma[7/10],10,120][[1]] (* Harvey P. Dale, Mar 02 2023 *)

gamma(.7) \\ Charles R Greathouse IV, Nov 26 2024

this * A340723 = Pi/A019863 [DLMF (5.5.3)]

this * A175380 * 2^(9/10)/sqrt(2*Pi) = 2*A246745. [DLMF (5.5.5)]

A091545 First column sequence of the array (7,2)-Stirling2 A091747.

Original entry on oeis.org

1, 42, 5544, 1507968, 696681216, 489070213632, 485157651922944, 646229992361361408, 1112808046846264344576, 2405890997281623512973312, 6380422924790865556405223424, 20366309975932442856045473169408, 77025384328976498881563979526701056, 340606249502734078054275917467072069632

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also sixth column (m=5) sequence of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A008548, A034323, A091543, A091747, A175380, A246745.

a[n_] := 5^(2*n) * Pochhammer[1/5, n] * Pochhammer[2/5, n] / 2; Array[a, 15] (* Amiram Eldar, Sep 01 2025 *)

a(n) = Product_{j=0..n-1} ((5*j+2)*(5*j+1))/2, n>=1. From eq.12 of the Blasiak et al. reference with r=7, s=2, k=1.
a(n) = (5^(2*n))*risefac(1/5, n)*risefac(2/5, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac5(5*n-3)*fac5(5*n-4)/2, n>=1, with fac5(5*n-4)/2 = A034323(n) and fac5(5*n-3) = A008548(n) (5-factorials).
E.g.f.: (hypergeom([1/5, 2/5], [], 25*x)-1)/2.
a(n) = A091747(n, 2), n>=1.
D-finite with recurrence a(n) - (5*n-3)*(5*n-4)*a(n-1) = 0. - R. J. Mathar, Jul 27 2022
a(n) ~ Pi * (5/e)^(2*n) * n^(2*n-2/5) / (Gamma(1/5) * Gamma(2/5)). - Amiram Eldar, Sep 01 2025
a(n) ~ sqrt(Pi*(1 + sqrt(5))) * 5^(2*n + 1/4) * n^(2*n - 2/5) / (Gamma(1/10) * 2^(7/10) * exp(2*n)). - Vaclav Kotesovec, Sep 01 2025

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A084940 Heptagorials: n-th polygorial for k=7.

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A256191 Decimal expansion of Gamma(1/10).

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A340721 Decimal expansion of Gamma(3/5).

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A340725 Decimal expansion of Gamma(9/10).

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A340724 Decimal expansion of Gamma(7/10).

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A091545 First column sequence of the array (7,2)-Stirling2 A091747.

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