A100221 -id:A100221 - cp's OEIS frontend

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

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

A015002 q-factorial numbers for q=4.

Original entry on oeis.org

1, 1, 5, 105, 8925, 3043425, 4154275125, 22686496457625, 495586515116818125, 43304845277422684580625, 15136126045591163828042953125, 21161832960467051739150680807015625, 118345540457280742481284963098558216328125, 2647344887069536899904944217513732945696167890625

Offset: 0

Olivier Gérard

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

Cf. A015001, A015004, A015005, A015006, A015007, A015008, A015009, A015011.
Column q=4 of A069777.
Cf. A002450, A100221.

[n le 1 select 1 else (4^n-1)*Self(n-1)/3: n in [1..15]]; // Vincenzo Librandi, Oct 22 2012

RecurrenceTable[{a[1]==1, a[n]==((4^n - 1) * a[n-1])/3}, a, {n, 15}] (* Vincenzo Librandi, Oct 27 2012 *)
Table[QFactorial[n, 4], {n, 15}] (* Bruno Berselli, Aug 14 2013 *)

a(n) = Product_{k=1..n} (q^k - 1) / (q - 1) with q=4.
a(0) = 1, a(n) = (4^n-1)*a(n-1)/3. - Vincenzo Librandi, Oct 27 2012
From Amiram Eldar, Jul 05 2025: (Start)
a(n) = Product_{k=1..n} A002450(k).
a(n) ~ c * 2^(n*(n+1))/3^n, where c = A100221. (End)

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

A132020 Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

This is the limiting probability that a large random symmetric binary matrix is nonsingular (cf. A086812, A048651). In other words, equals Lim_{n->oo} A086812(n)/A006125(n+1).- H. Tracy Hall, Sep 07 2024

			0.41942244179510759770995610770297425223395323439266674908044991663177...

John E. Cremona and R. W. K. Odoni, Some density results for negative Pell equations; an application of graph theory, Journal of the London Mathematical Society s2-39:1 (1989), pp. 16-28.

Cf. A004171, A048651, A098844, A067080, A132019, A132026, A132028, A100221, A000217, A081845.

evalf(1+sum((-1)^n*2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 23 2020

RealDigits[ Product[1 - 1/(2*4^i), {i, 0, 175}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2, 1/4], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)

prodinf(k=0,1-1.>>(2*k+1)) \\ Charles R Greathouse IV, Nov 16 2012

Equals lim inf_{n->oo} Product_{k=0..floor(log_4(n))} floor(n/4^k)*4^k/n.
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^((1/2)*(1+floor(log_4(n)))*floor(log_4(n))).
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^A000217(floor(log_4(n))).
Equals (1/2)*exp(-Sum_{n>0} (4^(-n)*(Sum_{k|n} 1/(k*2^k)))).
Equals lim inf_{n->oo} A132028(n)/A132028(n+1).
Equals Product_{k>0} (1-1/(2^k+1)). - Robert G. Wilson v, May 25 2011
From Robert FERREOL, Feb 23 2020: (Start)
Equals Product_{k>0} (1 + 1/2^k)^(-1) = 2/A081845.
Equals 1 + Sum_{n>=1} (-1)^n*2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). (End)
From Peter Bala, Jan 15 2021: (Start)
Constant C = Sum_{n >= 0} 2^n/Product_{k = 1..n} (1 - 4^k).
Faster converging series:
2*C = (1/2)*Sum_{n >= 0} 2^(-n)/Product_{k = 1..n} (1 - 4^k);
(2^4)*C = 7*Sum_{n >= 0} 2^(-3*n)/Product_{k = 1..n} (1 - 4^k);
(2^9)*C = 7*31*Sum_{n >= 0} 2^(-5*n)/Product_{k = 1..n} (1 - 4^k), and so on.
Slower converging series:
C = -Sum_{n >= 0} 2^(3*n)/Product_{k = 1..n} (1 - 4^k);
7*C = Sum_{n >= 0} 2^(5*n)/Product_{k = 1..n} (1 - 4^k);
7*31*C = -Sum_{n >= 0} 2^(7*n)/Product_{k = 1..n} (1 - 4^k), and so on. (End)
Equals Product_{n>=0} (1 - 1/A004171(n)). - Amiram Eldar, May 09 2023

Name corrected by Charles R Greathouse IV, Nov 16 2012

A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

Original entry on oeis.org

2, 48, 23040, 185794560, 24257337753600, 50821645356918374400, 1704875112338069448032256000, 915241991059360703024740763172864000, 7861748876453505095791592854589753555681280000, 1080506416218846625176535970968094253434513802154475520000, 2376056471052200653607636735377527394627947719754523173734842368000000

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). This is the sequence obtained by (illegally) setting p = 2.

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

Cf. A001309, A003956, A100221.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7).
A bisection of A003053, cf. A003923.

Table[2^(n^2+2n+1) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, May 14 2022 *)

from math import prod
def A090770(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) ~ c * 2^((n+1)*(2*n+1)), where c = A100221. - Amiram Eldar, Jul 06 2025

A003788 Order of universal Chevalley group A_n (4).

Original entry on oeis.org

1, 60, 60480, 987033600, 258492255436800, 1083930404878024704000, 72736898347485916060188672000, 78099458182389588115529148326215680000, 1341733356588640095264385107865053233298800640000

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.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Index entries for sequences related to groups.

Cf. A003787, A003789, A003790, A003791, A003792, A053291, A100221.

[&*[(4^n - 4^k): k in [0..n-1]]/3: n in [1..8]]; // Vincenzo Librandi, Sep 19 2015

f[m_, n_] := m^(n (n + 1)/2) Product[m^k - 1, {k, 2, n + 1}];
f[4, #] & /@ Range[0, 8] (* Michael De Vlieger, Sep 18 2015 *)

Numbers so far appear to equal A053291(n)/3. - Ralf Stephan, Mar 30 2004
a(n) = A(4,n) where A(q,n) is defined in A003787. - Sean A. Irvine, Sep 18 2015
a(n) ~ c * 4^(n*(n+2)), where c = (4/3) * A100221 = 0.918050049493... . - Amiram Eldar, Jul 07 2025

One more term from Sean A. Irvine, Sep 18 2015

A003923 Order of universal Chevalley group B_n (2) or symplectic group Sp(2n,2).

Original entry on oeis.org

1, 6, 720, 1451520, 47377612800, 24815256521932800, 208114637736580743168000, 27930968965434591767112450048000, 59980383884075203672726385914533642240000, 2060902435720151186326095525680721766346957783040000, 1132992015386677099994486205757869431795095310094129168384000000

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.

T. D. Noe, Table of n, a(n) for n = 0..20
Bernhard Runge, On Siegel modular forms. Part I, J. Reine Angew. Math., 436 (1993), 57-85.
Index entries for sequences related to groups.

A bisection of A003053.

Cf. A003920, A100221.

for m from 0 to 50 do N:=2^(m^2)*mul( 4^i-1, i=1..m); lprint(N); od:

a[n_] := 2^(n^2)*Times@@(4^Range[n]-1);
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Aug 18 2022 *)

from math import prod
def A003923(n): return (1 << n**2)*prod((1 << i)-1 for i in range(2,2*n+1,2)) # Chai Wah Wu, Jun 20 2022

a(n) = B(2,n) where B(q,n) is defined in A003920. - Sean A. Irvine, Sep 22 2015

a(n) ~ c * 2^(n*(2*n+1)), where c = A100221. - Amiram Eldar, Jul 07 2025

Edited by N. J. A. Sloane, Dec 30 2008

A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 512, 528, 544, 560, 648, 666, 684, 702, 800, 820, 840, 860, 968, 990, 1012, 1034, 1728, 1764, 1800, 1836, 2028, 2067, 2106, 2145, 2352

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base-4 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m-2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(26)=floor(26/4^0)*floor(26/4^1)*floor(26/4^2)=26*6*1=156; a(34)=544 since 34=202(base-4) and so
a(34)=202*20*2(base-4)=34*8*2=544.

Cf. A048651, A132020, A100221, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

Recurrence: a(n)=n*a(floor(n/4)); a(n*4^m)=n^m*4^(m(m+1)/2)*a(n).

a(k*4^m)=k^(m+1)*4^(m(m+1)/2), for 0

Asymptotic behavior: a(n)=O(n^((1+log_4(n))/2)); this follows from the inequalities below.

a(n)<=b(n), where b(n)=n^(1+floor(log_4(n)))/4^((1+floor(log_4(n)))*floor(log_4(n))/2); equality holds for n=k*4^m, 0=0. b(n) can also be written n^(1+floor(log_4(n)))/4^A000217(floor(log_4(n))).

Also: a(n)<=2^(1/4)*n^((1+log_4(n))/2)=1.189207...*4^A000217(log_4(n)), equality holds for n=2*4^m, m>=0.

a(n)>c*b(n), where c=0.4194224417951075977... (see constant A132020).

Also: a(n)>c*2^(1/4)*n^((1+log_4(n))/2)=0.498780...*4^A000217(log_4(n)).

lim inf a(n)/b(n)=0.4194224417951075977..., for n-->oo.

lim sup a(n)/b(n)=1, for n-->oo.

lim inf a(n)/n^((1+log_4(n))/2)=0.4194224417951075977...*2^(1/4), for n-->oo.

lim sup a(n)/n^((1+log_4(n))/2)=2^(1/4), for n-->oo.

lim inf a(n)/a(n+1)=0.4194224417951075977... for n-->oo (see constant A132020).

A014115 Order of a certain Clifford group in dimension 2^n (the automorphism group of the Barnes-Wall lattice for n != 3).

Original entry on oeis.org

2, 8, 1152, 2580480, 89181388800, 48126558103142400, 409825748158189771161600, 55428899652335313894424707072000

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.

A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. E. Wall, On the Clifford collineation, transform and similarity groups. IV. An application to quadratic forms, Nagoya Math. J., 21 (1962), pp. 199-222.
Index entries for sequences related to Barnes-Wall lattices.

Agrees with A014116 except at n=3.

Cf. A001309, A003956, A100221.

2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);

a[n_] := 2^(n^2+n+1)*(2^n - 1) * Product[4^i - 1, {i, 1, n-1}]; a[0] = 2; Array[a, 8, 0] (* Amiram Eldar, Jul 07 2025 *)

from math import prod
def A014115(n): return 2 if n == 0 else ((1<Chai Wah Wu, Jun 20 2022

From Amiram Eldar, Jul 07 2025: (Start)
a(n) = 2^(n^2+n+1) * (2^n-1) * Product_{i=1..n-1} (2^(2*i)-1).
a(n) ~ c * 2^(2*n^2+n+1), where c = A100221. (End)

A027638 Order of 2^n X 2^n unitary group H_n acting on Siegel modular forms.

Original entry on oeis.org

4, 96, 46080, 371589120, 48514675507200, 101643290713836748800, 3409750224676138896064512000, 1830483982118721406049481526345728000, 15723497752907010191583185709179507111362560000

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..39
Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
Bernhard Runge, On Siegel modular forms. Part I, J. Reine Angew. Math., 436 (1993), 57-85.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Index entries for sequences related to modular groups.

Cf. A003956, A027672, A027639, A100221.

A027638:= func< n | n eq 0 select 4 else 2^(n^2+2*n+2)*(&*[4^j-1: j in [1..n]]) >;
[A027638(n): n in [0..15]]; // G. C. Greubel, Aug 04 2022

seq( 2^(n^2+2*n+2)*product(4^i -1, i=1..n), n=0..12);

Table[2^(n^2+2n+2) Product[4^k-1,{k,n}],{n,0,10}] (* Harvey P. Dale, May 21 2018 *)

a(n) = my(ret=1); for(i=1,n, ret = ret<<(2*i)-ret); ret << (n^2+2*n+2); \\ Kevin Ryde, Aug 13 2022

from sage.combinat.q_analogues import q_pochhammer
def A027638(n): return (-1)^n*2^(n^2 + 2*n + 2)*q_pochhammer(n, 4, 4)
[A027638(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

a(n) = A003956(n)/2.
a(n) = 2^(n^2 + 2*n + 2) * Product_{j=1..n} (4^j - 1).
a(n) ~ c * 2^(2*n^2+3*n+2), where c = A100221. - Amiram Eldar, Jul 06 2025

A258459 Number of partitions of n into parts of exactly 4 sorts which are introduced in ascending order.

Original entry on oeis.org

1, 11, 77, 438, 2216, 10422, 46731, 202814, 860586, 3593561, 14834956, 60735095, 247155292, 1001318246, 4043482110, 16288762319, 65500024027, 263035832734, 1055252430510, 4230340216034, 16949359882259, 67881449170593, 271777855641517, 1087867649157513

Offset: 4

Alois P. Heinz, May 30 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A256130.

Cf. A320546.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
a:= n-> T(n,4):
seq(a(n), n=4..35);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i/(i!*(k - i)!), {i, 0, k}];
Table[T[n, 4], {n, 4, 35}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

a(n) ~ c * 4^n, where c = 1/(24*Product_{n>=1} (1-1/4^n)) = 1/(24*QPochhammer[1/4, 1/4]) = 1/(24*A100221) = 0.060514735102066542326446... . - Vaclav Kotesovec, Jun 01 2015

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cp's OEIS Frontend

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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A015002 q-factorial numbers for q=4.

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A132020 Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).

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A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

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A003788 Order of universal Chevalley group A_n (4).

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A003923 Order of universal Chevalley group B_n (2) or symplectic group Sp(2n,2).

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A132028 Product{0<=k<=floor(log_4(n)), floor(n/4^k)}, n>=1.

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