A100222 -id:A100222 - cp's OEIS frontend

A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).

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

Offset: 0

Eric W. Weisstein, Nov 09 2004

Limit of the probability that a random N X N matrix, with entries chosen independently and uniformly from the field F_3, is nonsingular [Morrison (2006)]. - L. Edson Jeffery, Jan 22 2012

			0.56012607792794894496979224331414001437973633379836...

G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A048651, A027871.

Cf. A000203, A100221, A100222, A132019, A132034, A132035, A132036, A132037, A132038, A258458.

N[(3^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[3]]^(1/3))/2^(1/3)]
N[QPochhammer[1/3,1/3]] (* G. C. Greubel, Nov 27 2015 *)

exp(-Sum_{k > 0} sigma_1(k)/k/3^k) = exp(-Sum_{k > 0} A000203(k)/k/3^k). - Hieronymus Fischer, Aug 07 2007
Product_{k >= 1} (1 - 1/3^k) = (1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
From Peter Bala, Jan 18 2021: (Start)
Constant C = (1 - 1/3)*Sum_{n >= 0} (-1/3)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*Sum_{n >= 0} (-1/9)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*(1 - 1/27)*Sum_{n >= 0} (-1/27)^n/Product_{k = 1..n} (3^k - 1), and so on. (End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(3)) * exp(log(3)/24 - Pi^2/(6*log(3))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027871(n). (End)

A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 09 2004

			0.68853753712033971545651435729350818467554981937833...

G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A027637, A048651, A100220, A100222.

Cf. A000203, A132020, A132034, A132035, A132036, A132037, A132038, A258459.

EllipticThetaPrime[1, 0, 1/2]^(1/3)/2^(1/4)
N[QPochhammer[1/4]] (* G. C. Greubel, Nov 30 2015 *)
RealDigits[Fold[Times,1-1/4^Range[1000]],10,110][[1]] (* Harvey P. Dale, Sep 27 2024 *)

prodinf(x=1, 1-1/4^x) \\ Altug Alkan, Dec 01 2015

Equals exp(-Sum_{k>0} sigma_1(k)/(k*4^k)) where sigma_1() is A000203(). - Hieronymus Fischer, Aug 07 2007
Equals (1/4; 1/4){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(Pi/log(2)) * exp(log(2)/12 - Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027637(n). (End)

A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99812906992595851327996232224527387813073843536581646175407814028299858...

Istvan Mezo, Several special values of Jacobi theta functions arXiv:1106.2703 [math.CA], 2011-2013
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292821, A292825, A292829.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-2Pi]], 10, 104] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-2*Pi)) = exp(Pi/12)*Gamma(1/4)/(2*Pi^(3/4)).

A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.954918789987674103751233978110291077632715373807805283148799191676094...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292820, A292824, A292828.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi]], 10, 104] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-Pi)) = exp(Pi/24)*Gamma(1/4)/(2^(7/8)*Pi^(3/4)).
Equals 1/exp(A255695). - Hugo Pfoertner, May 28 2025

A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.99999651264548223429509891685211924765750978932634584844773269100472...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A000706, A292822, A292826.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-4*Pi]], 10, 103] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-4*Pi)) = exp(Pi/6)*Gamma(1/4)/(2^(11/8)*Pi^(3/4)).
A259150 = A259148 * exp(Pi/8)/sqrt(2). - Vaclav Kotesovec, Jul 03 2017

A259151 Decimal expansion of phi(exp(-8*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.999999999987838443290442788140998270959486945673852198543872725583699...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-8*Pi]], 10, 103] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-8*Pi)) = (sqrt(2) - 1)^(1/4)*exp(Pi/3)*(Gamma(1/4)/(2^(29/16)*Pi^(3/4))).
A259151 = A259147 * exp(5*Pi/16)/2. - Vaclav Kotesovec, Jul 03 2017

A027872 a(n) = Product_{i=1..n} (5^i - 1).

Original entry on oeis.org

1, 4, 96, 11904, 7428096, 23205371904, 362560730628096, 28324694519589371904, 11064305472020078810628096, 21609960560733744406929189371904, 211034749490954911990173458030810628096

Offset: 0

N. J. A. Sloane

nonn

Given probability p = 1/5^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1 - a(n)/A109345(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100222 ~ 0.2396672. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A109345, A100222.

[1] cat [&*[ 5^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027872 := proc(n)
        mul( 5^i-1, i=1..n) ;
end proc: # R. J. Mathar, Mar 12 2013

Table[Product[(5^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[5, 5, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Join[{1},FoldList[Times,5^Range[10]-1]] (* Harvey P. Dale, Dec 28 2021 *)

a(n) = prod(i=1, n, 5^i-1); \\ Michel Marcus, Nov 21 2015

4^n|a(n) for n >= 1. - G. C. Greubel, Nov 21 2015
a(n) ~ c * 5^(n*(n+1)/2), where c = Product_{k>=1} (1-1/5^k) = A100222 . - Vaclav Kotesovec, Nov 21 2015
a(n) = 5^(binomial(n+1,2))*(1/5; 1/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A024049(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 5^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 5^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A100222. - Amiram Eldar, May 07 2023

A053292 Number of nonsingular n X n matrices over GF(5).

Original entry on oeis.org

1, 4, 480, 1488000, 116064000000, 226614960000000000, 11064475422000000000000000, 13506266841692625000000000000000000, 412177498341354683437500000000000000000000000

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..35
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=5 of A316622.

Cf. A002884, A003789, A027872, A053290, A053291, A053293, A109345, A100222.

[1] cat [&*[(5^n - 5^k): k in [0..n-1]]: n in [1..8]]; // Bruno Berselli, Jan 28 2013

Table[Product[5^n - 5^k, {k,0,n-1}], {n,0,10}] (* Geoffrey Critzer, Jan 26 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 5^n - 5^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = (5^n - 1)*(5^n - 5)*...*(5^n - 5^(n-1)).
a(n) = A109345(n)*A027872(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 5^(n^2), where c = A100222. - Amiram Eldar, Jul 06 2025

More terms from Vladeta Jovovic, Mar 16 2000

A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jun 19 2015

			0.74931147780000278742962565878338031190409252790117392831206731...

Istvan Mezo, Several special values of Jacobi theta functions, arXiv:1106.2703 [math.CA], 2011-2013.
Eric Weisstein's MathWorld, Infinite Product
Eric Weisstein's MathWorld, Jacobi Theta Functions
Eric Weisstein's MathWorld, q-Pochhammer Symbol
Wikipedia, Euler function

Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A259151 phi(exp(-8*Pi)), A363019 phi(exp(-10*Pi)), A363020 phi(exp(-12*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).

Cf. A292819, A292823, A292827.

phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi/2]], 10, 105] // First

phi(q) = QPochhammer(q,q) = (q;q)_infinity.
phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
phi(exp(-Pi/2)) = ((sqrt(2) - 1)^(1/3)*(4 + 3*sqrt(2))^(1/24) * exp(Pi/48) * Gamma(1/4))/(2^(5/6)*Pi^(3/4)).
phi(exp(-Pi/2)) = (sqrt(2)-1)^(1/4) * exp(Pi/48) * Gamma(1/4)/(2^(13/16)*Pi^(3/4)). - Vaclav Kotesovec, Jul 03 2017

A015004 q-factorial numbers for q=5.

Original entry on oeis.org

1, 1, 6, 186, 29016, 22661496, 88515803376, 1728802155736656, 168827903320618878336, 82435457461295106532780416, 201258420458750640859769304304896, 2456767777551003294245070550498298923776, 149949204558598784020761783280669552547300269056

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Index entries for sequences related to factorial numbers.

Column q=5 of A069777.
Cf. A015001, A015002, A015005, A015006, A015007, A015008, A015009, A015011.
Cf. A003463, A100222.

[n le 1 select 1 else (5^n-1)*Self(n-1)/4: n in [1..15]]; // Vincenzo Librandi, Oct 25 2012

RecurrenceTable[{a[1]==1, a[n]==((5^n - 1)* a[n-1])/4}, a, {n, 15}] (* Vincenzo Librandi, Oct 25 2012 *)
Table[QFactorial[n, 5], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = { my(q=5); prod(k=1, n, ((q^k - 1) / (q - 1))) } \\ Andrew Howroyd, Feb 18 2018

a(n) = Product_{k=1..n} ((q^k - 1) / (q - 1)) where q = 5.
a(0) = 1, a(n) = (5^n-1)*a(n-1)/4. - Vincenzo Librandi, Oct 25 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A003463(k).
a(n) ~ c * 5^(n*(n+1)/2)/4^n, where c = A100222. (End)

a(0)=1 prepended by Alois P. Heinz, Sep 08 2021

cp's OEIS Frontend

A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).

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A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).

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A259149 Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A259148 Decimal expansion of phi(exp(-Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A259150 Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A259151 Decimal expansion of phi(exp(-8*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

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A027872 a(n) = Product_{i=1..n} (5^i - 1).

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A053292 Number of nonsingular n X n matrices over GF(5).

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