A132037 -id:A132037 - cp's OEIS frontend

A028671 Pseudo Galois numbers for d=9; order of group AGL(n,3^2).

1, 72, 466560, 247608990720, 10657130578027315200, 37158487365982254056334950400, 10494634615565778355427184150449750016000, 240083527795435700509596514439839216948307004751872000, 444879613680905841995130298273091805272041653799250478127267184640000

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..32

Cf. A028668, A028670, A028672, A028674, A028676, A028677, A028678, A028680, A028682, A028683, A028684, A028686.

Cf. A132037.

FoldList[ #1*9^#2 (9^#2-1)&, 1, Range[ 20 ] ]
a[n_] := 9^n * Product[9^n - 9^k, {k, 0, n-1}]; Array[a, 9, 0] (* Amiram Eldar, Jul 13 2025 *)

a(n) = 9^n * prod(k = 0, n-1, 9^n - 9^k); \\ Amiram Eldar, Jul 13 2025

a(n) = 9^n * Product_{k=0..n-1} (9^n - 9^k).

a(n) ~ c * 9^(n^2+n), where c = A132037. - Amiram Eldar, Jul 13 2025

A003830 Order of universal Chevalley group D_n (3).

Original entry on oeis.org

2, 576, 12130560, 19808719257600, 2579025599882610278400, 27051378802435080953011843891200, 22941271269626791484963824552883153534976000, 1574947942338058195342953134725345263180893951172280320000

Offset: 1

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Robert Israel, Table of n, a(n) for n = 1..32
Robert Steinberg, Lectures on Chevalley Groups, Dept. of Mathematics, Yale University, 1967, pp. 130-131.
Index entries for sequences related to groups.

Cf. A003831, A003832, A003834, A003835, A003836, A132037.

f:= n -> 3^(n*(n-1))*(3^n-1)*mul(3^(2*k)-1,k=1..n-1):
map(f, [$1..10]); # Robert Israel, Sep 22 2015

f[m_, n_] := m^(n (n - 1)) (m^n - 1) Product[m^(2 k) - 1, {k, n - 1}];
f[3, #] & /@ Range@ 8 (* Michael De Vlieger, Sep 17 2015 *)

a(n,q=3) = q^(n*(n-1)) * (q^n-1) * prod(k=1,n-1,q^(2*k)-1); \\ Michel Marcus, Sep 17 2015

a(n) = D(3,n) where D(q,n) = q^(n*(n-1)) * (q^n-1) * Product_{k=1..n-1}(q^(2*k)-1). - Sean A. Irvine, Sep 17 2015

a(n) ~ c * 3^(n*(2*n-1)), where c = A132037. - Amiram Eldar, Jul 07 2025

a(8) and formula from Sean A. Irvine, Sep 17 2015

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

A015008 q-factorial numbers for q=9.

Original entry on oeis.org

1, 1, 10, 910, 746200, 5507702200, 365876657146000, 218747042884536166000, 1177042838234827583459440000, 57001313848230245122464621625840000, 24843911488189148287648216529610193612000000, 97453533413342456299179976631323547842824103012000000

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..40
Index entries for sequences related to factorial numbers.

Cf. A015001, A015002, A015004, A015005, A015006, A015007, A015009, A015011.
Column q=9 of A069777.
Cf. A002452, A132037.

[n le 1 select 1 else (9^n - 1)*Self(n-1)/8: n in [1..15]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==((9^n - 1) * a[n-1])/8}, a, {n, 15}] (* Vincenzo Librandi, Oct 26 2012 *)
Table[QFactorial[n, 9], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (9^k - 1) / (9 - 1).
a(0) = 1, a(n) = (9^n - 1)*a(n-1)/8. - Vincenzo Librandi, Oct 26 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A002452(k).
a(n) ~ c * 3^(n*(n+1))/8^n, where c = A132037. (End)

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

A003927 Order of simple Chevalley group B_n (3).

Original entry on oeis.org

25920, 4585351680, 65784756654489600, 76457792934119864313446400, 7197966128645938515382156481789952000, 54888780931741129517511777421088069718405808128000

Offset: 2

N. J. A. Sloane

nonn

Index entries for sequences related to groups.

Cf. A003920, A003938, A003939, A003940, A003941, A003942, A132037.

b[q_, n_] := q^(n^2) * Product[q^(2*k) - 1, {k, 1, n}] / GCD[2, q-1]; Table[b[3, n], {n, 2, 7}] (* Amiram Eldar, Jun 23 2025 *)

a(n) = b(3,n) where b(q,n) = q^(n^2) * Product_{k=1..n}(q^(2*k)-1) / gcd(2, q-1). - Sean A. Irvine, Sep 22 2015
a(n) = A003920(n) / 2. - Amiram Eldar, Jun 23 2025
a(n) ~ c * 3^(2*n^2+n), where c = A132037. - Amiram Eldar, Jul 09 2025

A003792 Order of universal Chevalley group A_n (9).

Original entry on oeis.org

1, 720, 42456960, 203039372390400, 78660280796419613491200, 2468438315722201136962330755072000, 6274437692242927471137606015213542491815936000, 1291851049702792234730057308758464452124128263449062932480000

Offset: 0

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

Index entries for sequences related to groups.

Cf. A003787, A003788, A003789, A003790, A003791, A052497, A132037.

[&*[(9^n - 9^k): k in [0..n-1]]/8: n in [1..10]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[9, #] & /@ Range[0, 7] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A052497(n)/8. - Ralf Stephan, Mar 30 2004
a(n) = A(9,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 9^(n*(n+2)), where c = (9/8) * A132037 = 0.9861303982904... . - Amiram Eldar, Jul 07 2025

a(7) from Sean A. Irvine, Sep 18 2015

A052497 Number of nonsingular n X n matrices over GF(9).

Original entry on oeis.org

1, 8, 5760, 339655680, 1624314979123200, 629282246371356907929600, 19747506525777609095698646040576000, 50195501537943419769100848121708339934527488000

Offset: 0

Vladeta Jovovic, Mar 16 2000

G. C. Greubel, Table of n, a(n) for n = 0..30
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.

Cf. A027877, A053764.

Cf. A002884, A003792, A053290, A053291, A053292, A053293, A052496, A052498, A132037.

[1] cat [&*[(9^n - 9^k): k in [0..n-1]]: n in [1..10]]; // Bruno Berselli, Jan 28 2013

Table[Product[(9^n - 9^j), {j, 0, n-1}], {n, 0, 10}] (* G. C. Greubel, May 14 2019 *)

{a(n) = prod(j=0,n-1, 9^n - 9^j)}; \\ G. C. Greubel, May 14 2019

[product(9^n - 9^j for j in (0..n-1)) for n in (0..10)] # G. C. Greubel, May 14 2019

a(n) = (9^n - 1)*(9^n - 9)*...*(9^n - 9^(n-1)).
a(n) = A053764(n)*A027877(n). - Bruno Berselli, Jan 30 2013
a(n) ~ c * 9^(n^2), where c = A132037. - Amiram Eldar, Jul 06 2025

A092299 a(n) = 43^(n^2+2n+1)Product_{j=1..n} (9^j-1).

Original entry on oeis.org

12, 2592, 50388480, 80225312993280, 10358730921842550374400, 108354149159204252828272715366400, 91807063616969429053277006948134413139968000, 6300752103463414524173850924959140409591369032708128768000, 35026261744325078751960598643637064012678383486922588643915999981076480000

Offset: 0

N. J. A. Sloane, Feb 10 2004

nonn

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A001309, A003956, A132037.

Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

a[n_] := 4*3^(n^2+2*n+1) * Product[9^j - 1, {j, 1, n}]; Array[a, 10, 0] (* Amiram Eldar, Jul 06 2025 *)

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 4 * A092301(n).
a(n) ~ c * 3^(2*n^2+3*n+1), where c = 4 * A132037. (End)

A092301 a(n) = 3^(n^2+2n+1)*Product_{j=1..n} (9^j-1).

Original entry on oeis.org

3, 648, 12597120, 20056328248320, 2589682730460637593600, 27088537289801063207068178841600, 22951765904242357263319251737033603284992000, 1575188025865853631043462731239785102397842258177032192000, 8756565436081269687990149660909266003169595871730647160978999995269120000

Offset: 0

N. J. A. Sloane, Feb 10 2004

nonn

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A001309, A003956, A132037.

Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7), A090770 (p=2, although this is the wrong formula in that case).

Table[3^(n^2+2n+1) Product[9^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Jun 23 2013 *)

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = A092299(n) / 4.
a(n) ~ c * 3^(2*n^2+3*n+1), where c = A132037. (End)

A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.4689451783670236932832800...

Cf. A048651, A098844, A067080, A132019, A132026, A132033, A132037, A000217, A270369.

digits = 103; NProduct[1-1/(2*9^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/9], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_9(n))} floor(n/9^k)*9^k/n.
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^(1/2*(1+floor(log_9(n)))*floor(log_9(n))).
Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^A000217(floor(log_9(n))).
Equals (1/2)*exp(-Sum_{n>0} 9^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132033(n)/A132033(n+1).
Equals Product_{n>=1} (1 - 1/A270369(n)). - Amiram Eldar, May 08 2023

cp's OEIS Frontend

A028671 Pseudo Galois numbers for d=9; order of group AGL(n,3^2).

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A003830 Order of universal Chevalley group D_n (3).

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Maple

Mathematica

PARI

Formula

Extensions

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

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Mathematica

Formula

A015008 q-factorial numbers for q=9.

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Magma

Mathematica

Formula

Extensions

A003927 Order of simple Chevalley group B_n (3).

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Mathematica

Formula

A003792 Order of universal Chevalley group A_n (9).

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Magma

Mathematica

Formula

Extensions

A052497 Number of nonsingular n X n matrices over GF(9).

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Magma

Mathematica

PARI

Sage

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A092299 a(n) = 4*3^(n^2+2n+1)*Product_{j=1..n} (9^j-1).

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A092299 a(n) = 43^(n^2+2n+1)Product_{j=1..n} (9^j-1).