A083864 -id:A083864 - cp's OEIS frontend

A081845 Decimal expansion of Product_{k>=0} (1 + 1/2^k).

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

Offset: 1

Benoit Cloitre, Apr 09 2003

Twice the product in A079555.

			4.76846205806274344829979857....

G. C. Greubel, Table of n, a(n) for n = 1..2000
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; citeseerx.
József Sándor, On Vandiver's arithmetical function - I, Notes on Number Theory and Discrete Mathematics, Vol. 27, No. 3 (2021), pp. 29-38. See p. 36, eq. 40.

Cf. A006125, A020696, A028361, A079555, A083864.

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

digits = 105; NProduct[1 + 1/2^k, {k, 0, Infinity}, WorkingPrecision -> digits+5, NProductFactors -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 04 2013 *)
N[QPochhammer[-1, 1/2], 100] (* Vaclav Kotesovec, Dec 13 2015 *)
2*N[QPochhammer[-1/2, 1/2], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(k=0,1/2^k,1) \\ Hugo Pfoertner, Feb 21 2020

lim sup Product_{k=0..floor(log_2(n))} (1 + 1/floor(n/2^k)) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132369(n)/A098844(n) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132269(n)/n^((1+log_2(n))/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
lim sup A132270(n)/n^((log_2(n)-1)/2) for n-->oo. - Hieronymus Fischer, Aug 20 2007
2*exp(Sum_{n>0} 2^(-n)*Sum_{k|n} -(-1)^k/k) = 2*exp(Sum_{n>0} A000593(n)/(n*2^n)). - Hieronymus Fischer, Aug 20 2007
lim sup A132269(n+1)/A132269(n) = 4.76846205806274344... for n-->oo. - Hieronymus Fischer, Aug 20 2007
Sum_{k>=1} (-1)^(k+1) * 2^k / (k*(2^k-1)) = log(A081845) = 1.562023833218500307570359922772014353168080202860122... . - Vaclav Kotesovec, Dec 13 2015
Equals 2*(-1/2; 1/2){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
Equals 1 + Sum_{n>=1} 2^n/((2-1)*(2^2-1)*...*(2^n-1)). - Robert FERREOL, Feb 21 2020
From Peter Bala, Jan 18 2021: (Start)
Constant C = 3*Sum_{n >= 0} (1/2)^n/Product_{k = 1..n} (2^k - 1).
Faster converging series:
C = (2*3*5)/(2^3)*Sum_{n >= 0} (1/4)^n/Product_{k = 1..n} (2^k - 1),
C = (2*3*5*9)/(2^6)*Sum_{n >= 0} (1/8)^n/Product_{k = 1..n} (2^k - 1),
C = (2*3*5*9*17)/(2^10)*Sum_{n >= 0} (1/16)^n/Product_{k = 1..n} (2^k - 1), and so on. The sequence [2,3,5,9,17,...] is A000051. (End)
From Amiram Eldar, Mar 20 2022: (Start)
Equals sqrt(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/A083864. (End)
Equals lim_{n->oo} A020696(2^n)/A006125(n+1) (Sándor, 2021). - Amiram Eldar, Jun 29 2022

A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).

Original entry on oeis.org

1, 4, 12, 36, 92, 228, 540, 1236, 2748, 6004, 12876, 27252, 57036, 118308, 243564, 498564, 1015484, 2060484, 4167804, 8409588, 16934748, 34049940, 68378220, 137185428, 275026476, 551052676, 1103618508, 2209525092, 4422484764, 8850120420, 17707920924

Offset: 0

Vaclav Kotesovec, Aug 25 2015

nonn

Cf. A032302, A070933, A261563.

nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1)), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 - QPochhammer[-2, x]/(3 QPochhammer[2, x]))[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = c * 2^n, where c = 1/(A048651 * A083864) = 2*Product_{j>=1} (2^j+1)/(2^j-1) = 16.5119758715565001310882816988645462530540032335764606912075051272567456...

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

A085011 Decimal expansion of 4*Product_{k>=0} (1 - 1/(2^k+1)).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jun 19 2003

			0.8388448835902...

Cf. A083864.

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

4*prodinf(k=0, 1-1/(2^k+1)) \\ Amiram Eldar, May 28 2021

Equals 4 * A083864. - Amiram Eldar, May 28 2021

Leading zero removed by R. J. Mathar, Feb 05 2009

A348875 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(2*x)).

Original entry on oeis.org

1, 2, 8, 56, 656, 13184, 477248, 32524928, 4295916032, 1117098857984, 576442191401984, 592587279827787776, 1215991461595100598272, 4985567391504232291377152, 40861715233637664786276712448, 669641809249948891254213657460736, 21945501536426419427607885034600595456

Offset: 0

Ilya Gutkovskiy, Nov 02 2021

nonn

Cf. A006318, A015083, A348876, A348877.

nmax = 16; A[] = 0; Do[A[x] = 1/(1 - x - x A[2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[2^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).

a(n) ~ c * 2^(n*(n-1)/2), where c = 1/(A048651 * A083864) = 2*Product_{j>=1} (2^j+1)/(2^j-1) = 16.51197587155650013108828169886454625305400323357646... - Vaclav Kotesovec, Nov 03 2021

A371749 Decimal expansion of Product_{k>=0} 1 / (1 + 1/4^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 05 2024

			0.368756127076900562750845672280819915482...

Cf. A000302, A083864, A100221, A273413, A371746, A371747, A371748, A371752.

RealDigits[1/QPochhammer[-1, 1/4], 10, 100][[1]]

Equals A273413^2. - Hugo Pfoertner, Apr 05 2024

A371752 Decimal expansion of Product_{k>=0} 1 / (1 + 1/5^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 05 2024

			0.39666773502905896079231190779208218756...

Cf. A000351, A083864, A100222, A371746, A371749, A371750, A371751.

RealDigits[1/QPochhammer[-1, 1/5], 10, 100][[1]]

A371746 Decimal expansion of Product_{k>=0} 1 / (1 + 1/3^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 05 2024

			0.31950228831873889019480071011090065424...

Cf. A000244, A083864, A100220, A132323, A370466, A371749, A371752.

RealDigits[1/QPochhammer[-1, 1/3], 10, 100][[1]]

Equals 1 / A132323.

A370398 Triangle read by rows: T(n,k) is the numerator of the probability of winning a 1-player game M(n,k) as defined below while playing optimally.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 11, 5, 1, 0, 4, 31, 11, 9, 1, 0, 4, 1, 69, 21, 19, 1, 0, 4, 101, 49, 829, 94, 34, 1, 0, 4, 2783, 3733, 56069, 25367, 551, 69, 1, 0, 32, 13439, 21517, 389573, 1543163, 14011, 565, 125, 1, 0, 32, 149621, 271643, 5709959, 379562191, 2757715, 1392901, 19388, 251, 1

Offset: 1

Julian Zbigniew Kuryllowicz-Kazmierczak, Feb 17 2024

A game M(n,k) is played as follows: play begins with k blue tokens and n-k red; 0 <= k <= n > 0. At each move, the order of the r remaining tokens is randomized and observed, and the player then chooses a number d, 1 <= d <= r/2, and discards the last d of the remaining tokens. The game is won iff the last remaining token is blue.
Let Pr(n,k) be the probability of winning a game M(n,k). Then Pr(n+1,1) = (n/(n-1))*Pr(n,1) if n is a power of 2, Pr(n,1) otherwise. So lim_{n->oo} Pr(n,1) = (1/2)*(2/3)*(4/5)*(8/9)*(16/17)*... = A083864.
The (generally suboptimal) strategy of removing 1 token on each move (regardless of the randomization result) provides Pr(n,k) >= k/n as a lower bound.

			The values of Pr(n,k) begin as follows:
.
  n\k|  0    1       2       3        4        5       6     7
  ---+---------------------------------------------------------
  1  | 0/1  1/1
  2  | 0/1  1/2     1/1
  3  | 0/1  1/3     2/3     1/1
  4  | 0/1  1/3    11/18    5/6      1/1
  5  | 0/1  4/15   31/60   11/15     9/10     1/1
  6  | 0/1  4/15    1/2    69/100   21/25    19/20    1/1
  7  | 0/1  4/15  101/210  49/75   829/1050  94/105  34/35  1/1
  ...
We can calculate Pr(4,2) given the values of Pr(n,k) for n=3 and n=2 as seen in the table below. The leftmost column lists each of the six possible outcomes (i.e., C(4,2) = 6 combinations, all equally likely) of randomizing the n=4 tokens during the first move; in each randomized sequence (i.e., combination), the red and blue tokens are represented by "r" and "b", respectively. Removing the last j = 1 or 2 tokens will leave n' = n - j remaining tokens of which k' = k - f(c,j) are blue. For each randomized sequence, an asterisk marks the probability of winning using the optimal choice of the number j of tokens to remove.
.
  randomized     if remove          if remove       probability
   sequence    last j=1 token    last j=2 tokens      of win
      of       ---------------   ---------------   given optimal
    tokens     n' k' Pr(n',k')   n' k' Pr(n',k')      choice
  ==========   == == =========   == == =========   =============
     rrbb       3  1    1/3 *     2  0    0/1           1/3
     rbrb       3  1    1/3       2  1    1/2 *         1/2
     brrb       3  1    1/3       2  1    1/2 *         1/2
     rbbr       3  2    2/3 *     2  1    1/2           2/3
     brbr       3  2    2/3 *     2  1    1/2           2/3
     bbrr       3  2    2/3       2  2    1/1 *         1/1
.
For example, when we get rbrb it's better to remove the last two tokens (one r and one b) instead of removing only the last token (b).
The probability of winning M(4,2) is the average of the probabilities of winning for each randomized sequence, i.e.,
Pr(4,2) = (1/3 + 1/2 + 1/2 + 2/3 + 2/3 + 1/1)/6 = 11/18.

Denominators are in A370399.

Cf. A083864.

T(n,k) = numerator(Pr(n,k)) where Pr(n,k) =
   0 if k = 0,
   1 if k = n, and
   (1/C(n,k))*Sum_{c=1..C(n,k)} Max_{j=1..floor(n/2)} Pr(n-j, k-f(c,j)) otherwise,
and where the summation is over all combinations of n tokens, exactly k of which are blue, and f(c,j) is the number of blue tokens among the last j tokens in the c-th combination.

More terms from Jon E. Schoenfield, Feb 24 2024

A370399 Triangle read by rows: T(n, k) is the denominator of the probability of winning a certain game while playing optimally.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 18, 6, 1, 1, 15, 60, 15, 10, 1, 1, 15, 2, 100, 25, 20, 1, 1, 15, 210, 75, 1050, 105, 35, 1, 1, 15, 5880, 5880, 73500, 29400, 588, 70, 1, 1, 135, 30240, 35280, 529200, 1852200, 15435, 588, 126, 1, 1, 135, 340200, 453600, 7938000, 466754400, 3111696, 1481760, 19845, 252, 1

Offset: 1

Julian Zbigniew Kuryllowicz-Kazmierczak, Feb 17 2024

T(n, k) is the numerator of the probability of winning a game M(n, k) while playing optimally. The game is played by a single player who begins with n tokens of which k are blue and the rest are red; 0 <= k <= n > 0. Each move consists of randomizing the order of the remaining tokens, observing their resulting order, choosing a number m of tokens to remove, and removing the m tokens that are ordered last; m must be at least 1 but no more than half of the remaining tokens. Play continues until only one token remains; the game is won if that token is blue, otherwise the game is lost.

Let Pr(n,k) be the probability of winning a game M(n,k). Then Pr(n+1,1) = (n/(n-1))*Pr(n,1) if n is a power of 2, Pr(n,1) otherwise. So lim_{n->oo} Pr(n,1) = (1/2)*(2/3)*(4/5)*(8/9)*(16/17)*... = A083864.

			The values of Pr(n,k) begin as follows:
.
  n\k|  0    1       2       3        4        5       6     7
  ---+---------------------------------------------------------
  1  | 0/1  1/1
  2  | 0/1  1/2     1/1
  3  | 0/1  1/3     2/3     1/1
  4  | 0/1  1/3    11/18    5/6      1/1
  5  | 0/1  4/15   31/60   11/15     9/10     1/1
  6  | 0/1  4/15    1/2    69/100   21/25    19/20    1/1
  7  | 0/1  4/15  101/210  49/75   829/1050  94/105  34/35  1/1
  ...
We can calculate Pr(4,2) using the table below, given the values of Pr(n,k) for n=3 and for n=2. The leftmost column lists each of the six possible results of randomizing the n=4 tokens during the first move; in each randomized sequence, the red and blue tokens are represented by "r" and "b", respectively.
.
  randomized  probability    result if       result if
   sequence       of       last 1 token  last 2 tokens
  of tokens   occurrence     is removed     are removed
  ==========  ===========  ==============  =============
     rrbb         1/6      Pr(3,1) = 1/3   Pr(2,0) = 0/1
     rbrb         1/6      Pr(3,1) = 1/3   Pr(2,1) = 1/2
     brrb         1/6      Pr(3,1) = 1/3   Pr(2,1) = 1/2
     rbbr         1/6      Pr(3,2) = 2/3   Pr(2,1) = 1/2
     brbr         1/6      Pr(3,2) = 2/3   Pr(2,1) = 1/2
     bbrr         1/6      Pr(3,2) = 2/3   Pr(2,2) = 1/1
.
For example, when we get rbrb it's better to remove the last two tokens (one r and one b) instead of removing only the last token (b). So the probability of winning M(4,2) is
Pr(4,2) = (1/6)(1/3) + (1/6)(1/2) + (1/6)(1/2) + (1/6)(2/3) + (1/6)(2/3) + (1/6)(1/1) = 11/18.
Of course Pr(n,k) >= k/n, because k/n could be achieved by removing 1 token on each move.

Numerators are in A370398.

More terms from Jon E. Schoenfield, Feb 24 2024

cp's OEIS Frontend

A081845 Decimal expansion of Product_{k>=0} (1 + 1/2^k).

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A261584 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - 2*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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A085011 Decimal expansion of 4*Product_{k>=0} (1 - 1/(2^k+1)).

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A348875 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(2*x)).

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A371749 Decimal expansion of Product_{k>=0} 1 / (1 + 1/4^k).

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A371752 Decimal expansion of Product_{k>=0} 1 / (1 + 1/5^k).

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A371746 Decimal expansion of Product_{k>=0} 1 / (1 + 1/3^k).

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A370398 Triangle read by rows: T(n,k) is the numerator of the probability of winning a 1-player game M(n,k) as defined below while playing optimally.

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A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).

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