A261584 -id:A261584 - cp's OEIS frontend

A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k).

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

Offset: 0

N. J. A. Sloane

This is the limiting probability that a large random binary matrix is nonsingular (cf. A002884).
This constant is very close to 2^(13/24) * sqrt(Pi/log(2)) / exp(Pi^2/(6*log(2))) = 0.288788095086602421278899775042039398383022429351580356839... - Vaclav Kotesovec, Aug 21 2018
This constant is irrational (see Penn link). - Paolo Xausa, Dec 09 2024

			(1/2)*(3/4)*(7/8)*(15/16)*... = 0.288788095086602421278899721929230780088911904840685784114741...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 318, 354-361.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Steven R. Finch, Digital Search Tree Constants. [Broken link]
Steven R. Finch, Digital Search Tree Constants. [From the Wayback machine]
Marvin Geiselhart, Ahmed Elkelesh, Moustafa Ebada, Sebastian Cammerer, and Stephan ten Brink, On the Automorphism Group of Polar Codes, arXiv:2101.09679 [cs.IT], 2021.
T. S. Jayram, Hellinger strikes back: a note on the multi-party information complexity of AND, LNCS 5687 (2009) 562-573.
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.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Michael Penn, A non-standard irrationality proof, YouTube video, 2024.
V. Arvind Rameshwar, Shreyas Jain, and Navin Kashyap, Sampling-Based Estimates of the Sizes of Constrained Subcodes of Reed-Muller Codes, arXiv:2309.08907 [cs.IT], 2023.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, arXiv preprint arXiv:1608.00795 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Tree Searching.
Wikipedia, Pentagonal number theorem.
Wanchen Zhang, Yu Ning, Fei Shi, and Xiande Zhang, Extremal Maximal Entanglement, arXiv:2411.12208 [quant-ph], 2024. See p. 15.

Cf. A002884, A001318, A005327, A005329, A048652, A065446, A079555, A098844, A067080, A100220, A132019, A132020, A132026, A132038, A070933, A261584, A335764.

RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2], 10, 100][[1]] (* Jean-François Alcover, Nov 18 2015 *)

default(realprecision, 20080); x=prodinf(k=1, -1/2^k, 1); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b048651.txt", n, " ", d)); \\ Harry J. Smith, May 07 2009

exp(-Sum_{k>0} sigma_1(k)/k*2^(-k)) = exp(-Sum_{k>0} A000203(k)/k*2^(-k)). - Hieronymus Fischer, Jul 28 2007
From Hieronymus Fischer, Aug 13 2007: (Start)
Equals lim inf Product_{k=0..floor(log_2(n))} floor(n/2^k)*2^k/n for n->oo.
Equals lim inf A098844(n)/n^(1+floor(log_2(n)))*2^(1/2*(1+floor(log_2(n)))*floor(log_2(n))) for n->oo.
Equals lim inf A098844(n)/n^(1+floor(log_2(n)))*2^A000217(floor(log_2(n))) for n->oo.
Equals lim inf A098844(n)/(n+1)^((1+log_2(n+1))/2) for n->oo.
Equals (1/2)*exp(-Sum_{n>0} 2^(-n)*Sum_{k|n} 1/(k*2^k)). (End)
Limit of A177510(n)/A000079(n-1) as n->infinity (conjecture). - Mats Granvik, Mar 27 2011
Product_{k >= 1} (1-1/2^k) = (1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
exp(Sum_{n>=1}(1/n/(1 - 2^n))) (according to Mathematica). - Mats Granvik, Sep 07 2016
(Sum_{k>0} (4^k-1)/(Product_{i=1..k} ((4^i-1)*(2*4^i-1))))*2 = 2/7 + 2/(3*7*31) + 2/(3*7*15*31*127)+2/(3*7*15*31*63*127*511) + ... (conjecture). - Werner Schulte, Dec 22 2016
Equals Sum_{k=-oo..oo} (-1)^k/2^((3*k+1)*k/2) (by Euler's pentagonal number theorem). - Amiram Eldar, Aug 13 2020
From Peter Bala, Dec 15 2020: (Start)
Constant C = Sum_{n >= 0} (-1)^n/( Product_{k = 1..n} (2^k - 1) ). The above conjectural result by Schulte follows by adding terms of this series in pairs.
C = (1/2)*Sum_{n >= 0} (-1/2)^n/( Product_{k = 1..n} (2^k - 1) ).
C = (3/8)*Sum_{n >= 0} (-1/4)^n/( Product_{k = 1..n} (2^k - 1) ).
1/C = Sum_{n >= 0} 2^(n*(n-1)/2)/( Product_{k = 1..n} (2^k - 1) ).
C = 1 - Sum_{n >= 0} (1/2)^(n+1)*Product_{k = 1..n} (1 - 1/2^k).
This latter identity generalizes as:
C = Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*C = 1 - Sum_{n >= 0} (1/8)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*C = 6 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
3*7*15*C = 91 - Sum_{n >= 0} (1/32)^(n+1)*Product_{k = 1..n} (1 - 1/2^k),
and so on, where the sequence [1, 0, 1, 6, 91, ...] is A005327.
(End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(2)) * exp(log(2)/24 - Pi^2/(6*log(2))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A005329(n).
Equals exp(-A335764). (End)
Equals 1/A065446. - Hugo Pfoertner, Nov 23 2024

Corrected by Hieronymus Fischer, Jul 28 2007

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)

A265758 Expansion of Product_{k>=1} ((1 + kx^k)/(1 - kx^k)).

Original entry on oeis.org

1, 2, 6, 16, 38, 88, 200, 428, 902, 1874, 3780, 7504, 14732, 28368, 54052, 101960, 189750, 349996, 640218, 1159624, 2084952, 3722008, 6593560, 11606268, 20308188, 35312170, 61065636, 105060200, 179795936, 306244136, 519291476, 876554860, 1473504846

Offset: 0

Vaclav Kotesovec, Dec 15 2015

nonn

Convolution of A022629 and A006906.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A006906, A015128, A022629, A022661, A022693, A261584.

nmax = 40; CoefficientList[Series[Product[(1 + k*x^k)/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^(n/3), where
c = 28711548.45004804552683870974706458425598... if mod(n,3) = 0
c = 28711547.74098394497470795294574937283075... if mod(n,3) = 1
c = 28711547.58138731567204220029302329316039... if mod(n,3) = 2.

A303346 Expansion of Product_{n>=1} ((1 + 2x^n)/(1 - 2x^n))^(1/2).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 72, 142, 260, 510, 940, 1814, 3362, 6490, 12112, 23466, 44114, 85766, 162516, 317190, 604806, 1184682, 2271248, 4461514, 8591784, 16916490, 32696708, 64496130, 125037142, 247007142, 480077432, 949510526, 1849375796, 3661330398, 7144215452

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A032302, A070933, A261584, A303307, A303345, A370709, A370713.

nmax = 30; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[Sqrt[-QPochhammer[-2, x] / (3*QPochhammer[2, x])], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+2*x^k)/(1-2*x^k))^(1/2)))

a(n) ~ 2^n / sqrt(c*Pi*n), where c = A048651 * A083864 = 1/2 * Product_{j>=1} (2^j-1)/(2^j+1) = 0.06056210400129025123042464659093375290492912341... - Vaclav Kotesovec, Apr 22 2018

A264686 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k).

Original entry on oeis.org

1, 3, 6, 15, 27, 51, 93, 159, 264, 432, 696, 1086, 1683, 2553, 3837, 5700, 8367, 12147, 17505, 24972, 35361, 49728, 69402, 96243, 132657, 181782, 247692, 335838, 453042, 608289, 813102, 1082256, 1434519, 1894215, 2491644, 3265869, 4265973, 5553771, 7207167

Offset: 0

Vaclav Kotesovec, Nov 21 2015

nonn

Convolution of A000041 and A032302.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A006951, A015128, A032302, A261584, A264685, A266821.

Column k=3 of A321884.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 2*b(n-i, i-1))))
    end:
a:= n-> add(b(i$2)*combinat[numbpart](n-i), i=0..n):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 22 2017

nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

{ my(n=40); Vec(prod(k=1, n, 3/(1-x^k) - 2 + O(x*x^n))) } \\ Andrew Howroyd, Dec 22 2017

a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(3)*n), where c = 2*Pi^2/3 + log(2)^2 + 2*polylog(2, -1/2) = 6.163360867463814765670634381079217086937812673723341... . - Vaclav Kotesovec, Jan 04 2016

A261563 Expansion of Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^k.

Original entry on oeis.org

1, 4, 16, 60, 192, 596, 1744, 4892, 13248, 34868, 89296, 223660, 548928, 1323060, 3137520, 7332332, 16907584, 38517444, 86777328, 193523404, 427562816, 936555044, 2035286576, 4390850268, 9409096576, 20037827876, 42429318480, 89369282460, 187325508288

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A156616, A261561, A261562, A261584.

nmax = 50; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1))^2, {k, 1, nmax}]], {x, 0, nmax}], x]

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, (2^d - (-2)^d) * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Sep 30 2015

a(n) ~ c * 2^n, where c = 2 * Product_{j>=1} ((1 + 1/2^j)/(1 - 1/2^j))^(j+1) = 1021.5383556752320172813996404366861329314041364322798995039038143325883...

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} (2^d - (-2)^d) * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

Original entry on oeis.org

1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..1600

Cf. A261568, A246936, A067855, A303350.

Cf. A015128, A261584, A303390.

N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019

nmax = 25; CoefficientList[Series[Product[(1+4*x^k)/(1-4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = QPochhammer[-1, 1/4] / QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

Original entry on oeis.org

1, 3, 9, 24, 60, 141, 324, 717, 1560, 3330, 7020, 14622, 30225, 61998, 126522, 257007, 520326, 1050396, 2116116, 4255584, 8547330, 17149350, 34382295, 68889840, 137969466, 276220962, 552865365, 1106356314, 2213644548, 4428657402, 8859340926, 17721640698

Offset: 0

Vaclav Kotesovec, Nov 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A006951, A070933, A261584, A264686.

nmax = 40; CoefficientList[Series[Product[(1 + x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 2^n, where c = A079555 / A048651 = Product_{k>=1} (2^k+1)/(2^k-1) = 8.25598793577825006554414084943227312652...

A303390 Expansion of Product_{k>=1} (1 + 3x^k)/(1 - 3x^k).

Original entry on oeis.org

1, 6, 24, 96, 330, 1104, 3552, 11184, 34584, 105990, 322224, 975264, 2942016, 8857680, 26631312, 80005632, 240219114, 721036320, 2163789816, 6492625152, 19480105392, 58444390176, 175340344416, 526034008752, 1578124753152, 4734415061142, 14203316252400

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Cf. A032308, A242587.

Cf. A015128, A261584, A303391.

nmax = 30; CoefficientList[Series[Product[(1+3*x^k)/(1-3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where c = QPochhammer[-1, 1/3] / QPochhammer[1/3] = 5.5877920355220979147599292926505407983327527...

A303397 Expansion of Product_{k>=1} (1 - 2x^k)/(1 + 2x^k).

Original entry on oeis.org

1, -4, 4, -4, 20, -36, 52, -116, 244, -500, 964, -1876, 3876, -7780, 15332, -30628, 61684, -123460, 246036, -491988, 985492, -1971284, 3939556, -7878068, 15762692, -31527428, 63041220, -126078916, 252185044, -504375460, 1008698036, -2017385268, 4034873268

Offset: 0

Seiichi Manyama, Apr 23 2018

sign

Expansion of Product_{k>=1} (1 - b*x^k)/(1 + b*x^k): A002448 (b=1), this sequence (b=2), A303398 (b=3).

Cf. A261584, A303345.

nmax = 40; CoefficientList[Series[Product[(1 - 2*x^k)/(1 + 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-2*x^k)/(1+2*x^k)))

a(n) ~ c * (-2)^n, where c = QPochhammer[-1, -1/2]/QPochhammer[-1/2] = 0.93943604828296530723602398257349307281... - Vaclav Kotesovec, Apr 25 2018

cp's OEIS Frontend

A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k).

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

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A265758 Expansion of Product_{k>=1} ((1 + k*x^k)/(1 - k*x^k)).

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A303346 Expansion of Product_{n>=1} ((1 + 2*x^n)/(1 - 2*x^n))^(1/2).

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A264686 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k).

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A261563 Expansion of Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^k.

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A303391 Expansion of Product_{k>=1} (1 + 4*x^k)/(1 - 4*x^k).

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A264685 Expansion of Product_{k>=1} (1 + x^k)/(1 - 2*x^k).

A265758 Expansion of Product_{k>=1} ((1 + kx^k)/(1 - kx^k)).

A303346 Expansion of Product_{n>=1} ((1 + 2x^n)/(1 - 2x^n))^(1/2).

A261563 Expansion of Product_{k>=1} ((1 + 2x^k)/(1 - 2x^k))^k.

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

A303390 Expansion of Product_{k>=1} (1 + 3x^k)/(1 - 3x^k).

A303397 Expansion of Product_{k>=1} (1 - 2x^k)/(1 + 2x^k).