A256191 -id:A256191 - cp's OEIS frontend

A020018 Nearest integer to Gamma(n + 1/10)/Gamma(1/10).

1, 0, 0, 0, 1, 3, 15, 91, 649, 5253, 47802, 482796, 5359036, 64844333, 849460756, 11977396665, 180858689642, 2911824903230, 49792205845229, 901238925798638, 17213663482753993, 345994636003355265, 7300486819670796088

Offset: 0

Simon Plouffe

nonn

			Gamma(1/10)/Gamma(1/10) = 1, so a(0) = 1.
Gamma(1 + 1/10)/Gamma(1/10) = 1/10 < 1/2, so a(1) = 0.
Gamma(2 + 1/10)/Gamma(1/10) = 11/100 < 1/2, so a(2) = 0.
Gamma(3 + 1/10)/Gamma(1/10) = 231/1000 < 1/2, so a(3) = 0.
Gamma(4 + 1/10)/Gamma(1/10) = 7161/10000 = 0.7161, so a(4) = 1.
Gamma(5 + 1/10)/Gamma(1/10) = 293601/100000 = 2.93601, so a(5) = 3.
Gamma(6 + 1/10)/Gamma(1/10) = 14973651/1000000 = 14.973651, so a(6) = 15.

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A045757, A020063, A020108, A000007 (decimal expansion of 1/10), A256191 (decimal expansion of Gamma(1/10)).

[Round(Gamma(n +1/10)/Gamma(1/10)): n in [0..30]]; // G. C. Greubel, Jan 20 2018

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

Table[Round[Gamma[n + 1/10]/Gamma[1/10]], {n, 0, 50}] (* G. C. Greubel, Jan 20 2018 *)

for(n=0,30, print1(round(gamma(n+1/10)/gamma(1/10)), ", ")) \\ G. C. Greubel, Jan 20 2018

A371881 Decimal expansion of Gamma(1/20).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			19.4700853112555128640473209677271275456304195833419756810827837553645...

Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371983.

evalf(GAMMA(1/20), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/20], 10, 120][[1]]
RealDigits[2^(33/40) * 5^(5/16) * (1 + Sqrt[5])^(1/8) * Sqrt[5^(1/4) + Sqrt[2 + Sqrt[5]]] * Sqrt[Pi * Gamma[1/10]] * QPochhammer[E^(-2*Sqrt[5]*Pi)] / E^(Sqrt[5]*Pi/12), 10, 120][[1]]

Equals 2^(33/40) * 5^(5/16) * (1 + sqrt(5))^(1/8) * sqrt(5^(1/4) + sqrt(2 + sqrt(5))) * sqrt(Pi*Gamma(1/10)) * QPochhammer(exp(-2*sqrt(5)*Pi)) / exp(sqrt(5)*Pi/12).

A035323 Related to deca-factorial numbers A045757.

Original entry on oeis.org

1, 55, 3850, 298375, 24466750, 2079673750, 181228712500, 16084048234375, 1447564341093750, 131728355039531250, 12095058053629687500, 1118792869960746093750, 104133797896346367187500, 9743948231729552929687500, 915931133782577975390625000, 86441000750730796427490234375

Offset: 1

Wolfdieter Lang

Convolution of A035308(n-1) with A025755(n), n >= 1.

Michael De Vlieger, Table of n, a(n) for n = 1..502
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.
Index entries for sequences related to factorial numbers.

Cf. A045757, A035308, A025755, A256191.

Rest@ CoefficientList[Series[(-1 + (1 - 100 x)^(-1/10))/10, {x, 0, 13}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 10^(n-1)*A045757(n)/n!, where A045757(n) = (10*n-9)(!^10) = Product_{j=1..n} (10*j-9).
G.f.: (-1+(1-100*x)^(-1/10))/10.
D-finite with recurrence: n*a(n) + 10*(-10*n+9)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 10^(2*n-1) * n^(-9/10) / Gamma(1/10). - Amiram Eldar, Aug 18 2025

A371983 Decimal expansion of Gamma(1/30).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			29.4547797456996940196962082886383457347018736055729711046565415567498...

Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 2009.
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371881.

evalf(GAMMA(1/30), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/30], 10, 120][[1]]
RealDigits[2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma[1/5] * Gamma[1/3] / ((10 + Sqrt[5] - Sqrt[75 + 30*Sqrt[5]])^(1/4) * Sqrt[Pi]), 10, 120][[1]]

Equals 3^(9/20) * sqrt(5 + sqrt(5)) * sqrt(sqrt(15) + sqrt(5 + 2*sqrt(5))) * Gamma(1/3) * Gamma(1/5) / (sqrt(Pi) * 2^(16/15) * 5^(1/6)).
Equals 2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma(1/5) * Gamma(1/3) / ((10 + sqrt(5) - sqrt(75 + 30*sqrt(5)))^(1/4) * sqrt(Pi)).
Equals 8*Pi^2 / (Gamma(17/30) * Gamma(19/30) * Gamma(23/30)).
Equals Gamma(7/30) * Gamma(11/30) * Gamma(13/30) / (2*Pi*A019815).

A366316 Decimal expansion of Gamma(1/10) / (Gamma(2/15) * Gamma(7/15)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 06 2023

			0.71202391272664934751491237088911174403906744162700306692997...

Mathematics StackExchange, Is it possible to simplify Gamma(1/10) / (Gamma(2/15) * Gamma(7/15)) ?, 2013.
MathOverflow, When are some products of gamma functions algebraic numbers?, 2009.
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.

Cf. A256191.

RealDigits[Gamma[1/10]/Gamma[2/15]/Gamma[7/15], 10, 120][[1]]
RealDigits[(Sqrt[5] + 1) / (3^(1/10) * 2^(6/5) * Sqrt[Pi]), 10, 120][[1]]

Equals (1 + sqrt(5)) / (3^(1/10) * 2^(6/5) * sqrt(Pi)).

cp's OEIS Frontend

A020018 Nearest integer to Gamma(n + 1/10)/Gamma(1/10).

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A371881 Decimal expansion of Gamma(1/20).

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A035323 Related to deca-factorial numbers A045757.

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A371983 Decimal expansion of Gamma(1/30).

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A366316 Decimal expansion of Gamma(1/10) / (Gamma(2/15) * Gamma(7/15)).

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