A081845 -id:A081845 - cp's OEIS frontend

A028361 Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.

1, 2, 6, 30, 270, 4590, 151470, 9845550, 1270075950, 326409519150, 167448083323950, 171634285407048750, 351678650799042888750, 1440827432323678715208750, 11804699153027899713705288750, 193419995622362136809061156168750, 6338179836549184861096125026493768750

Offset: 0

N. J. A. Sloane

These numbers appear in first column of A155103. - Mats Granvik, Jan 20 2009
Equals row sums of unsigned triangle A158474. - Gary W. Adamson, Mar 20 2009
a(n) = (n+2) terms in the sequence (1, 1, 2, 4, 8, 16, ...) dot (n+2) terms in the sequence (1, 1, 2, 6, 30, 270, ...). Example: a(4) = 4590 = (1, 2, 4, 8, 16) dot (1, 1, 2, 6, 30, 270) = (1 + 1 + 4 + 24 + 240 + 4320). - Gary W. Adamson, Aug 02 2010
a(n) is the right-hand side of the mass formula used to classify Type II Self Dual Binary Linear Codes of length 2(n+1). a(n) is the number of distinct Type II Self Dual Binary Linear codes of length 2(n+1) when 2(n+1) = 0 MOD 8.  It is important to note that Type II codes are only possible when the length is a multiple of 8. In short, this sequence only applies to Type II codes when 2(n+1) = 0 MOD 8, else the right hand side of the mass formula is zero. - Nathan J. Russell, Mar 04 2016
This is almost certainly the sequence of number of Carlyle circles needed for the construction of regular polygons using straightedge and compass mentioned on page 107 of DeTemple (1991). - N. J. A. Sloane, Aug 05 2021
a(n) is also the number of Sp(oo, F2)-orbits of V^n, where V is the countable-dimensional symplectic vector space over the two-element field. - Jingjie Yang, Jul 30 2025

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Page 366. - Nathan J. Russell, Mar 04 2016

T. D. Noe, Table of n, a(n) for n=0..50
C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
Julien Clément and Antoine Genitrini, Binary Decision Diagrams: from Tree Compaction to Sampling, arXiv:1907.06743 [cs.DS], 2019.
John P. D'Angelo, Counting Tournament Brackets, J. Int. Seq., Vol. 25 (2022), Article 22.6.8.
Duane W. DeTemple, Carlyle circles and the Lemoine simplicity of polygon constructions, The American Mathematical Monthly 98.2 (1991): 97-108.
Davide Gaiotto and Justin Kulp, Orbifold groupoids, arXiv:2008.05960 [hep-th], 2020, page 42.

Cf. A006125, A028362, A155103, A158474, A323716 (product of 3^i+1).

[1] cat [ (&*[1+2^j: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Jun 06 2020

seq( mul((1+2^j), j=0..n-1), n = 0..20); # G. C. Greubel, Jun 06 2020

Table[QPochhammer[-1, 2, n], {n, 0, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *)
Table[Product[2^i + 1, {i, 0, n/2 - 2}], {n, 2, 32, 2}] (* Nathan J. Russell, Mar 04 2016 *)
Table[Product[2^i + 1, {i, 0, n - 1}], {n, 0, 15}] (* Nathan J. Russell, Mar 04 2016 *)
FoldList[Times,1,2^Range[0,20]+1] (* Harvey P. Dale, Apr 11 2016 *)

PARI
```
{a(n) = prod(k=0, n-1, 2^k + 1)};
```

{a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m,1-2^k*x+x*O(x^n))),n)} /* Paul D. Hanna, May 02 2012 */

for n in range(2,50,2):
  product = 1
  for i in range(0,n//2-2 + 1):
    product *= (2**i+1)
  print(product)
# Nathan J. Russell, Mar 01 2016

[product( 1+2^j for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Jun 06 2020

a(n) = Product_{i=0..n-1} ( 2^i + 1 ).
Asymptotic to C*2^(n*(n-1)/2) where C = A081845 = 4.76846205806274344829979857... = Product_{k>=0} (1 + 1/2^k). - Benoit Cloitre, Apr 09 2003
It appears that a(n) = 2^((1/2)*(n - 1)*n) * Product_{k>=0} (1 + 1/(2^k)) / Product_{k>=0} (1 + 1/(2^(n + k))). - Peter Moxey (pmoxey(AT)live.com), Mar 21 2010
G.f.: Sum_{n>=0} 2^(n*(n-1)/2) * x^n / Product_{k=0..n} (1 - 2^k*x). - Paul D. Hanna, May 02 2012
a(n) = (a(n-2)^3 + a(n-1) * a(n-3) * (a(n-1) - 2 * a(n-2))) * a(n-1) / (a(n-2)^2 * (a(n-2) - a(n-3))) if n>2. - Michael Somos, Aug 21 2012
0 = a(n)*(+a(n+1) + a(n+2)) + a(n+1)*(-2*a(n+1)) for all n>=0. - Michael Somos, Oct 10 2014
Sum_{k=0..n} 2^k/a(k) = 3-2/a(n) and Sum_{k=0..n} 4^k/a(k) = 9-(4*(1+2^n))/a(n) for n >= 0. - Werner Schulte, Dec 25 2016
G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
a(n) = Sum_{k=0..n} q_binomial(n, k, q=2) * 2^(k*(k-1)/2). - Jingjie Yang, Jul 30 2025

A079555 Decimal expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 25 2003

			2.38423102903137172414989928867839723877161951650843345769...

G. C. Greubel, Table of n, a(n) for n = 1..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.

Cf. A028362, A048651, A081845, A098844, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132269, A132270, A000593.

Cf. A141848. - Gleb Koloskov, Apr 04 2021

digits = 105; NProduct[(1 + 1/2^k), {k, 1, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> 200] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 14 2013 *)
N[QPochhammer[-1/2,1/2]] (* G. C. Greubel, Dec 05 2015 *)
1/N[QPochhammer[1/2, 1/4]] (* Gleb Koloskov, Apr 04 2021 *)

prodinf(n=1,1+2.^-n) \\ Charles R Greathouse IV, May 27 2015

1/prodinf(n=0, 1-2^(-2*n-1)) \\ Gleb Koloskov, Apr 04 2021

(1/2)*lim sup Product_{k=0..floor(log_2(n)), (1 + 1/floor(n/2^k))} for n-->oo. - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132369(n)/A098844(n) for n-->oo. - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132269(n)/n^((1+log_2(n))/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132270(n)/n^((log_2(n)-1)/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
exp(sum{n>0, 2^(-n)*sum{k|n, -(-1)^k/k}})=exp(sum{n>0, A000593(n)/(n*2^n)}). - Hieronymus Fischer, Aug 20 2007
(1/2)*lim sup A132269(n+1)/A132269(n)=2.3842310290313717241498992886... for n-->oo. - Hieronymus Fischer, Aug 20 2007
Equals (-1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 05 2015
2 + Sum_{k>1} 1/(Product_{i=2..k} (2^i-1)) = 2 + 1/3 + 1/(3*7) + 1/(3*7*15) + 1/(3*7*15*31) + 1/(3*7*15*31*63) + ... (conjecture). - Werner Schulte, Dec 22 2016
From Peter Bala, Dec 15 2020: (Start)
The above conjecture of Schulte follows by setting x = 1/2 and t = -1 in the identity Product_{k >= 1} (1 - t*x^k) = Sum_{n >= 0} (-1)^n*x^(n*(n+1)/2)*t^n/( Product_{k = 1..n} 1 - x^k ), due to Euler.
Constant C = 1 + Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
C = 2 + Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
3*C = 7 + Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
3*7*C = 50 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
3*7*15*C = 751 + Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 + 1/2^k).
(End)
Equals 1/(1-P), where P is the Pell constant from A141848. - Gleb Koloskov, Apr 04 2021
Equals Sum_{k>=0} A000009(k)/2^k. - Vaclav Kotesovec, Sep 15 2021
From Amiram Eldar, Feb 19 2022: (Start)
Equals (sqrt(2)/2) * exp(log(2)/24 + Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(2))) (McIntosh, 1995).
Equals (1/2) * A081845.
Equals Sum_{n>=0} 1/A005329(n). (End)

A132269 a(n) = Product_{k>=0} (1 + floor(n/2^k)).

Original entry on oeis.org

1, 2, 6, 8, 30, 36, 56, 64, 270, 300, 396, 432, 728, 784, 960, 1024, 4590, 4860, 5700, 6000, 8316, 8712, 9936, 10368, 18200, 18928, 21168, 21952, 27840, 28800, 31744, 32768, 151470, 156060, 170100, 174960, 210900, 216600, 234000, 240000, 340956, 349272, 374616

Offset: 0

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base 2 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 (1+d(m)d(m-1)d(m-2)...d(2)d(1)d(0))*(1+d(m)d(m-1)d(m-2)...d(2)d(1))*(1+d(m)d(m-1)d(m-2)...d(2))*...*(1+d(m)d(m-1)d(m-2))*(1+d(m)d(m-1))*(1+d(m)).
From Gary W. Adamson, Aug 25 2016: (Start)
Given the following production matrix M =
  1, 0, 0, 0, 0, ...
  2, 0, 0, 0, 0, ...
  0, 3, 0, 0, 0, ...
  0, 4, 0, 0, 0, ...
  0, 0, 5, 0, 0, ...
  0, 0, 6, 0, 0, ...
  0, 0, 0, 7, 0, ...
  ...
the sequence is the left-shifted vector as lim_{n->infinity} M^n. (End)

			a(10) = (1 + floor(10/2^0))*(1 + floor(10/2^1))*(1 + floor(10/2^2))*(1 + floor(10/2^3)) = 11*6*3*2 = 396;
a(17) = 4860 since 17 = 10001_2 and so a(17) = (1+10001_2)*(1+1000_2)*(1+100_2)*(1+10_2)*(1+1) = 18*9*5*3*2 = 4860.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A048651, A081845, A132270, A132271(p=10), A132272, A132327(p=3), A132328.
For formulas regarding a general parameter p (i.e., terms 1+floor(n/p^k)) see A132271.
For the product of terms floor(n/p^k) see A098844, A067080, A132027-A132033, A132263, A132264.

[1] cat [n le 1 select 2 else (1+n)*Self(Floor(n/2)): n in [1..50]]; // Vincenzo Librandi, Aug 26 2016

f:= proc(n) option remember; (1+n)*procname(floor(n/2)) end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Aug 26 2016

Table[Product[1 + Floor[2 n/2^k], {k, 2 n}], {n, 0, 42}] (* or *)
Table[Function[w, Times @@ Map[1 + FromDigits[PadRight[w, #], 2] &, Range@ Length@ w]]@ IntegerDigits[n, 2], {n, 0, 42}] (* Michael De Vlieger, Aug 26 2016 *)

Recurrence: a(n)=(1+n)*a(floor(n/2)); a(2n)=(1+2n)*a(n); a(n*2^m) = (Product_{k=1..m} (1 + n*2^k))*a(n).
a(2^m-1) = 2^(m*(m+1)/2), a(2^m) = 2^(m*(m+1)/2)*Product_{k=0..m} (1 + 1/2^k), m>=1.
a(n) = A132270(2n) = (1+n)*A132270(n).
Asymptotic behavior: a(n) = O(n^((1+log_2(n))/2)); this follows from the inequalities below.
a(n) <= A098844(n)*Product_{k=0..floor(log_2(n))} (1 + 1/2^k).
a(n) >= A098844(n)/Product_{k=1..floor(log_2(n))} (1 - 1/2^k).
a(n) < c*n^((1+log_2(n))/2) = c*2^A000217(log_2(n)), where c = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627... (see constant A081845).
a(n) > n^((1+log_2(n))/2) = 2^A000217(log_2(n)),
lim sup a(n)/A098844(n) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627..., for n->oo (see constant A081845).
lim inf a(n)/A098844(n) = 1/Product_{k>=1} (1 - 1/2^k) = 1/0.288788095086602421..., for n->oo (see constant A048651).
lim inf a(n)/n^((1+log_2(n))/2) = 1, for n->oo.
lim sup a(n)/n^((1+log_2(n))/2) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627..., for n->oo (see constant A081845).
lim inf a(n+1)/a(n) = Product_{k>=0} (1 + 1/2^k) = 4.7684620580627... for n->oo (see constant A081845).
G.f. g(x) satisfies g(x) = (1+2x)*g(x^2) + 2*x^2*(1+x)*g'(x^2). - Robert Israel, Aug 26 2016

A132323 Decimal expansion of Product_{k>=0} (1+1/3^k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Aug 20 2007

Twice the constant A132324.

			3.12986803713402307587769821345767...

G. C. Greubel, Table of n, a(n) for n = 1..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.

Cf. A081845, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132324-A132326, A132327, A132328, A000593.

digits = 105; NProduct[1+1/3^k, {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
2*N[QPochhammer[-1/3,1/3]] (* G. C. Greubel, Dec 01 2015 *)

prodinf(x=0, 1+(1/3)^x) \\ Altug Alkan, Dec 03 2015

Equals lim sup_{n->oo} Product_{0<=k<=floor(log_3(n))} (1+1/floor(n/3^k)).
Equals lim sup_{n->oo} A132327(n)/A132027(n).
Equals lim sup_{n->oo} A132327(n)/n^((1+log_3(n))/2).
Equals lim sup_{n->oo} A132328(n)/n^((log_3(n)-1)/2).
Equals 2*exp(Sum_{n>0} 3^(-n) * Sum{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*3^n)).
Equals lim sup_{n->oo} A132327(n+1)/A132327(n).
Equals 2*(-1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015
Equals sqrt(2) * exp(log(3)/24 + Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(3))) (McIntosh, 1995). - Amiram Eldar, May 25 2023

A083864 Decimal expansion of Product_{k>=0} (1 - 1/(2^k+1)).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jun 19 2003

c/4 where c is the constant defined in A085011.

			0.2097112208975537988549780538514871...

Cf. A081845, A085011, A261584.

RealDigits[1/QPochhammer[-1, 1/2], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

PARI
```
prod(k=0,1000,1-1./(2^k+1))
```

prodinf(k=0, 1-1/(2^k+1)) \\ Michel Marcus, Feb 28 2020

Product_{k>=0} (1-1/(2^k+1)).
From Robert FERREOL, Feb 28 2020: (Start)
Equals Product_{k>=0} (1 + 1/2^k)^(-1) = 1/A081845.
Equals 1 + Sum_{k>=1} (-1)^k*2^(k*(k+1)/2)/((2-1)*(2^2-1)*...*(2^k-1)). (End)
From Peter Bala, Jan 16 2021: (Start)
Constant C = 2^(-1)*Sum_{n >= 0} (-1/2)^n/Product_{k = 1..n} (1 - 1/2^k).
C = (2^2/(3*5))*Sum_{n >= 0} (-1/8)^n/Product_{k = 1..n} (1 - 1/2^k).
C = (2^9/(3*5*9*17))*Sum_{n >= 0} (-1/32)^n/Product_{k = 1..n} (1 - 1/2^k).
C = (2^20/(3*5*9*17*33*65))*Sum_{n >= 0} (-1/128)^n/Product_{k = 1..n} (1 - 1/2^k) and so on. (End)

A126443 a(n) = Sum_{k=0..n-1} C(n-1,k)a(k)2^k for n>0, with a(0)=1.

Original entry on oeis.org

1, 1, 3, 17, 179, 3489, 127459, 8873137, 1195313043, 315321098561, 164239990789571, 169810102632595281, 349630019758589841523, 1436268949679165936016097, 11784559509424676876673518499, 193243076262167105764611875139569

Offset: 0

Paul D. Hanna, Jan 01 2007

nonn

Generated by a generalization of a recurrence for the Bell numbers (A000110).

Starting with offset 1 = eigensequence of triangle A013609. - Gary W. Adamson, Sep 04 2009

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A126347, A000110.
Cf. A013609. - Gary W. Adamson, Sep 04 2009
Column k=2 of A306245.

a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*a(k)*2^k))

a(n) = Sum_{k=0..n*(n-1)/2} A126347(n,k)*2^k.
G.f. A(x) satisfies: A(x) = 1 + x*A(2*x/(1 - x))/(1 - x). - Ilya Gutkovskiy, Sep 02 2019
a(n) ~ c * 2^(n*(n-1)/2), where c = A081845 = 4.7684620580627434482997985... - Vaclav Kotesovec, Sep 16 2019

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

A132325 Decimal expansion of Product_{k>=0} (1+1/10^k).

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, Aug 20 2007

Twice the constant A132326.

			2.22446913827410126425215613418881160749501...

G. C. Greubel, Table of n, a(n) for n = 1..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.

Cf. A081845, A067080, A100220, A132019-A132026, A132034-A132038, A132323-A132326, A132271, A132272, A000593.

digits = 105; NProduct[1+1/10^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
2*N[QPochhammer[-1/10,1/10]] (* G. C. Greubel, Dec 02 2015 *)

prodinf(x=0, 1+(1/10)^x) \\ Altug Alkan, Dec 03 2015

Equals lim sup_{n->oo} Product_{0<=k<=floor(log_10(n))} (1+1/floor(n/10^k)).
Equals lim sup_{n->oo} A132271(n)/n^((1+log_10(n))/2).
Equals lim sup_{n->oo} A132272(n)/n^((log_10(n)-1)/2).
Equals 2*exp(Sum_{n>0} 10^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*10^n)).
Equals lim sup_{n->oo} A132271(n+1)/A132271(n).
Equals 2*(-1/10; 1/10){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 02 2015
Equals sqrt(2) * exp(log(10)/24 + Pi^2/(12*log(10))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(10))) (McIntosh, 1995). - Amiram Eldar, May 20 2023

A006088 a(n) = (2^n + 2) a(n-1) (kissing number of Barnes-Wall lattice in dimension 2^n).

Original entry on oeis.org

1, 4, 24, 240, 4320, 146880, 9694080, 1260230400, 325139443200, 167121673804800, 171466837323724800, 351507016513635840000, 1440475753672879672320000, 11803258325595576034990080000, 193408190923209108909347450880000

Offset: 0

N. J. A. Sloane, John Leech

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 151.
Robert L. Griess Jr., Pieces of 2^d: Existence and uniqueness for Barnes-Wall and Ypsilanti lattices, arXiv:math/0403480 [math.GR], Mar 28 2004. See Proposition 8.9.
J. Leech, Some sphere packings in higher space, Canad. J. Math., 16 (1964), 657-682.
J. Leech & N. J. A. Sloane, Correspondence, 1975
C. Musès, The dimensional family approach in (hyper)sphere packing..., Applied Math. Computation 88 (1997), pp. 1-26, see p. 22.
G. Nebe and N. J. A. Sloane, Table of highest kissing numbers known
Index entries for sequences related to Barnes-Wall lattices

Cf. A028362. - Paul D. Hanna, Sep 16 2009

Cf. A081845.

I:=[4]; [1] cat [n le 1 select I[n] else (2^n + 2)*Self(n-1): n in [1..20]]; // Vincenzo Librandi, Dec 31 2015

a[0]:=1: for n from 1 to 16 do a[n]:=(2^n+2)*a[n-1] od: seq(a[n],n=0..16); # Emeric Deutsch, Dec 10 2004

RecurrenceTable[{a[0]==1, a[n]==(2^n + 2) a[n-1]}, a[n], {n, 0, 25}] (* Vincenzo Librandi, Dec 31 2015 *)

{a(n)=polcoeff(sum(m=0,n,2^(m*(m+1)/2)*x^m/prod(k=1,m+1,1-2^k*x+x*O(x^n))),n)} \\ Paul D. Hanna, Sep 16 2009

a(n) = prod(k=1, n, 2+2^k); \\ Michel Marcus, Jan 01 2016

a(n) = (2+2)(2+4)(2+8)(2+16)...(2+2^n).
From Paul D. Hanna, Sep 16 2009: (Start)
G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1-2^k*x)];
contrast with:
1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1+2^k*x)]. (End)
a(n) ~ c * 2^(n*(n+1)/2), where c = A081845. - Vaclav Kotesovec, Dec 31 2015

More terms from Emeric Deutsch, Dec 10 2004

Replaced arXiv URL with non-cached version - R. J. Mathar, Oct 23 2009

A330862 Decimal expansion of Product_{k>=1} (1 - 1/(-2)^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 28 2020

			(1 + 1/2) * (1 - 1/2^2) * (1 + 1/2^3) * (1 - 1/2^4) * (1 + 1/2^5) * ... = 1.2107241303010591801361728561...

Cf. A000203, A048651, A065446, A079555, A081845, A085011, A100221, A132020, A330863.

RealDigits[QPochhammer[-1/2, -1/2], 10, 111] [[1]]
N[QPochhammer[-2, 1/4]*QPochhammer[1/4]/3, 120] (* Vaclav Kotesovec, Apr 28 2020 *)

prodinf(k=1, 1 - 1/(-2)^k) \\ Michel Marcus, Apr 28 2020

Equals Product_{k>=1} (4^k - 1)*(4^k + 2)/4^(2*k).

Equals exp(-Sum_{k>=1} A000203(k)/(k*(-2)^k)).

cp's OEIS Frontend

A028361 Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.

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A079555 Decimal expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...

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A132269 a(n) = Product_{k>=0} (1 + floor(n/2^k)).

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A132323 Decimal expansion of Product_{k>=0} (1+1/3^k).

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A083864 Decimal expansion of Product_{k>=0} (1 - 1/(2^k+1)).

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A126443 a(n) = Sum_{k=0..n-1} C(n-1,k)*a(k)*2^k for n>0, with a(0)=1.

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

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A126443 a(n) = Sum_{k=0..n-1} C(n-1,k)a(k)2^k for n>0, with a(0)=1.