A186706 -id:A186706 - cp's OEIS frontend

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

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

Offset: 1

Paolo Xausa, Oct 11 2024

			5.2386596251856584103292320999763662681359773992...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A091925.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376859 (m = 4), A376911 (m = 5 and m = 6), A376912 (m = 7).

First[RealDigits[640*Pi^3/(2187*Sqrt[3]), 10, 100]]

Equals 640*Pi^3/(2187*sqrt(3)) = 640*A091925/(3^7*A002194) (cf. eq. 90 in Weisstein link).

A271923 Numerator of (1/3)(Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6j+1))) - 1).

Original entry on oeis.org

1, 5, 29, 52, 913, 1693, 69769, 658529, 1667651, 57873, 1616141, 1035959, 79918969, 3244922897, 3402714857, 6606018008, 51386679347, 5504537914811, 622652618545649, 10572475711004, 10931562934889, 235301799307039, 4608689892802861, 9034390134407023, 488936376609325, 959905250448181

Offset: 1

N. J. A. Sloane, May 04 2016

			1, 5/3, 29/13, 52/19, 913/285, 1693/465, 69769/17205, 658529/147963, 1667651/ 345247, 57873/11137, 1616141/291153, 1035959/175741, 79918969/12829093, ...

Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.

Sequences of fractions from de Gier paper: A271919-A271926.

Cf. A271924 (denominators), A073005, A186706.

f3:=proc(n) local j;
(1/3)*(mul(((2*j+1)*(3*j+4))/((j+1)*(6*j+1)),j=0..n-1)-1); end;
t3:=[seq(f3(n),n=1..50)];
map(numer,t3);
map(denom,t3);

a[n_] := (1/3)*(Product[((2*j + 1)*(3*j + 4))/((j + 1)*(6*j + 1)), {j, 0, n - 1}] - 1) // Numerator;
Array[a, 26] (* Jean-François Alcover, Nov 30 2017 *)

a(n)/A271924(n) ~ c * (2*n)^(2/3), where c = Gamma(1/3)*sqrt(3)/(2*Pi) = A073005/A186706. - Amiram Eldar, Aug 17 2025

A271925 Numerator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).

Original entry on oeis.org

3, 5, 87, 156, 913, 1693, 69769, 658529, 5002953, 173619, 1616141, 3107877, 239756907, 3244922897, 3402714857, 6606018008, 51386679347, 5504537914811, 622652618545649, 10572475711004, 10931562934889, 235301799307039, 4608689892802861, 9034390134407023, 488936376609325, 959905250448181

Offset: 1

N. J. A. Sloane, May 04 2016

			3, 5, 87/13, 156/19, 913/95, 1693/155, 69769/5735, 658529/49321, 5002953/345247, 173619/11137, 1616141/97051, 3107877/175741, 239756907/12829093, ...

Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.

Sequences of fractions from de Gier paper: A271919-A271926.

Cf. A271926 (denominators), A073005, A186706.

f3:=proc(n) local j;
(mul(((2*j+1)*(3*j+4))/((j+1)*(6*j+1)),j=0..n-1)-1); end;
t3:=[seq(f3(n),n=1..50)];
map(numer,t3);
map(denom,t3);

Table[Product[(2*j+1)*(3*j+4)/((j+1)*(6*j+1)),{j,0,n-1}]-1, {n,1,20}]//Numerator (* Vaclav Kotesovec, Oct 13 2017 *)

a(n)/A271926(n) ~ c * (2*n)^(2/3), where c = Gamma(1/3)*3^(3/2)/(2*Pi) = 3*A073005/A186706. - Amiram Eldar, Aug 17 2025

A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.61492662744600073515092236909361355359472818864859650504087875301429651...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.5.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), this sequence (Bi(0)), A358561 (Bi'(0)), A358564(Gi(0)).

RealDigits[AiryBi[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
airy(0)[2]
```
PARI
```
airy(0)[1]*sqrt(3)
```
PARI
```
3^(1/3)*gamma(1/3)/(2*Pi)
```

airy_bi(0).n(algorithm='scipy', prec=250)

Bi(0) = A284867*A002194.
Bi(0) = A358564*3.
Bi(0) = 1/(3^(1/6)*A073006).
Bi(0) = A073005/(3^(1/6)*A186706).
Bi(0) = A073005/(3^(1/6)*2*A093602).
Bi(0) = 3^(1/3)*A073005/(2*A000796).
Bi(0) = A252799/(3^(1/6)*BarnesG[5/3]).
Bi(0) = 3^(1/4)/(2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.44828835735382635791482371039882839086622679921226206108280877837233075...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.6.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), this sequence (Bi'(0)), A358564 (Gi(0)).

RealDigits[AiryBi'[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
derivnum(x=0, airy(x)[2])
```

airy_bi_prime(0).n(algorithm='scipy', prec=250)

Bi'(0) = A284868*A002194.
Bi'(0) = 3*Gi'(0), where Gi' is the derivative of the inhomogeneous Airy function of the first kind.
Bi'(0) = 3^(1/6)/A073005.
Bi'(0) = A073006*3^(1/6)/A186706.
Bi'(0) = A073006*3^(1/6)/2*A093602.
Bi'(0) = 3^(2/3)*A073006/(2*A000796).
Bi'(0) = 3^(1/4)*AGM(2,(sqrt(2+sqrt(3))))^(1/3)/(2^(7/9) * Pi^(2/3)), where AGM is the arithmetic-geometric mean.

A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.204975542482000245050307456364537851198242729549532168346959584338098839...

Scorer, R. S., Numerical evaluation of integrals of the form Integral_{x=x1..x2} f(x)*e^(i*phi(x))dx and the tabulation of the function Gi(z)=(1/Pi)*Integral_{u=0..oo} sin(u*z+u^3/3) du, Quart. J. Mech. Appl. Math. 3 (1950), 107-112.

Amparo Gil, Javier Segura, and Nico Temme, On nonoscillating integrals for computing inhomogeneous Airy functions, Mathematics of Computation 70.235 (2001): 1183-1194.
[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.12.6.
Allan J. MacLeod, Computation of inhomogeneous Airy functions, Journal of Computational and Applied Mathematics, Volume 53, Issue 1, 1994, Pages 109-116, ISSN 0377-0427.
Wikipedia, Scorer's function.
Index entries for transcendental numbers.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), A358561 (Bi'(0)), this sequence (Gi(0)).

First[RealDigits[N[ScorerGi[0],90]]] (* Stefano Spezia, Nov 28 2022 *)

PARI
```
airy(0)[2]/3
```
PARI
```
1/(3^(7/6)*gamma(2/3))
```
PARI
```
sqrt(3)*gamma(1/3)/(3^(7/6)*2*Pi)
```

1/(3^(3/4)*2^(2/9)*Pi^(1/3)*sqrtn(agm(2,(sqrt(2+sqrt(3)))),3))

1/(3^(7/6)*gamma(2/3)).n(algorithm='scipy', prec=250)

Gi(0) = A358559/3.
Gi(0) = A284867/A002194.
Gi(0) = Hi(0)/2, where Hi is the inhomogeneous Airy function of the second kind.
Gi(0) = 1/(3^(7/6)*A073006).
Gi(0) = A073005/(3^(7/6)*A186706).
Gi(0) = A073005/(3^(7/6)*2*A093602).
Gi(0) = A073005/(3^(4/6)*2*A000796).
Gi(0) = A252799/(3^(7/6)*BarnesG(5/3)).
Gi(0) = 1/(3^(3/4) * 2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A220610 Decimal expansion of sqrt(2*Pi^3).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 25 2012

This is the case n=4 of Product_{i=1..n-1} Gamma(i/n) = sqrt((2*Pi)^(n-1)/n).

Continued fraction expansion: 7, 1, 6, 1, 79, 4, 7, 1, 1, 1, 1, 1, 1, 4, 2, 3, 73, 1, 2, 1, 14, 3, 2, 1, 1, 2, 3, 1, ...

			7.8748049728612098721453229972336022711558426913993669291...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Particular values of the Gamma function: Products
Index entries for transcendental numbers

Cf. numbers of the form sqrt((2*Pi)^(n-1)/n) -- see the first comment: A002161 (n=2), A186706 (n=3).

SetDefaultRealField(RealField(100)); R:= RealField(); Sqrt(2*Pi(R)^3); // G. C. Greubel, Sep 29 2018

evalf(sqrt(2*Pi^3),120); # Muniru A Asiru, Sep 30 2018

Mathematica
```
RealDigits[Sqrt[2 Pi^3], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(sqrt(2*%pi^3)));
```

default(realprecision, 100); sqrt(2*Pi^3) \\ G. C. Greubel, Sep 29 2018

Equals A002193*A175476.

A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 6, 4, 0, 0, 6, 8, 6, 4, 0, 0, 0, 8, 12, 0, 0, 0, 6, 8, 0, 8, 0, 0, 6, 4, 12, 4, 0, 0, 0, 8, 6, 0, 0, 0, 0, 8, 12, 8, 0, 0, 12, 8, 0, 0, 0, 0, 0, 12, 6, 0, 0, 0, 6, 0, 12, 8, 0, 0, 0, 8, 12, 8, 0, 0, 0, 8, 0, 0, 0, 0, 6, 8, 12, 4, 0, 0, 12, 8, 0, 4, 0, 0

Offset: 0

Michael Somos, Jan 06 2015

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + 4*x + 6*x^2 + 4*x^3 + 6*x^6 + 8*x^7 + 6*x^8 + 4*x^9 + 8*x^13 + ...

Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004016, A033687, A033762, A097195, A112604, A112605, A121361, A123884, A186706, A244339, A253625.

Cf. A000122, A000700, A005882, A005928, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 83); A[1] + 4*A[2] + 6*A[3] + 4*A[4];

a[ n_] := SeriesCoefficient[ 1 + 2 Sum[ (1 + Mod[k, 2]) q^k / (1 - q^k + q^(2 k)), {k, n}], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 / (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^n Sum[(-1)^(n/d) {2, -1, 0, 1, -2, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, (n/d%2 + 1) * (-1)^(d\3) * (d%3>0) ))};

{a(n) = if( n<1, n==0, 2 * (-1)^n * sumdiv(n, d, (-1)^(n/d) * [0, 2, -1, 0, 1, -2][d%6 + 1]))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A)^4 * eta(x^12 + A) / (eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 4 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3/2*(e%2), if( p==3, 1, if( p%6 == 1, e+1, 1-e%2))))))};

Expansion of phi(q)^2 * phi(-q^3)^2 / (phi(-q^2) * phi(-q^6)) = psi(q) * psi(-q^3) * (chi(q) * chi(-q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of (2*a(q) + 3*a(q^2) - 2*a(q^4)) / 3 = (b(q) - 2*b(q^4)) * (b(q) - 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^2)^8 * eta(q^3)^4 * eta(q^12) / (eta(q)^4 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 4, -4, 0, -1, 4, -4, 4, -1, 0, -4, 4, -2, ...].
Moebius transform is period 12 sequence [ 4, 2, 0, -6, -4, 0, 4, 6, 0, -2, -4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253625.
a(n) = 4*b(n) where b() is multiplicative with b(2^e) = (3/4) * (1 - (-1)^e) if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: 1 + Sum_{k>0} (1 + (k mod 2)) * q^k / (1 - q^k + q^(2*k)).
G.f.: Product_{k>0} (1 + q^k) * (1 - q^(2*k)) * (1 - q^(3*k)) * (1 + q^(6*k)) / ((1 + q^(2*k)) * (1 - q^k + q^(2*k)))^3.
a(n) = (-1)^n * A244339(n). a(2*n) = A004016(n). a(2*n + 1) = 4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = 4 * A097195(n). a(6*n + 2) = 6 * A033687(n). a(6*n + 4) = a(6*n = 5) = 0.
a(12*n + 1) = 4 * A123884(n). a(12*n + 2) = 6 * A097195(n). a(12*n + 3) = 4 * A112604(n). a(12*n + 7) = 8 * A121361(n). a(12*n + 9) = 4 * A112605(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Dec 30 2023

cp's OEIS Frontend

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

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A271923 Numerator of (1/3)*(Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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A271925 Numerator of (Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

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A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

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A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

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A220610 Decimal expansion of sqrt(2*Pi^3).

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A271923 Numerator of (1/3)(Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6j+1))) - 1).

A271925 Numerator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).