A065446 -id:A065446 - cp's OEIS frontend

A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.

1, 1, 7, 155, 11811, 3309747, 3548836819, 14877590196755, 246614610741341843, 16256896431763117598611, 4274137206973266943778085267, 4488323837657412597958687922896275, 18839183877670041942218307147122500601235

Offset: 0

Paul D. Hanna, Oct 28 2012

nonn

Compare to: [x^n] Product_{k=0..n-1} 1+2^k*x = 2^(n*(n-1)/2).

			The coefficients in Product_{k=0..n-1} 1/(1 - 2^k*x) begin:
n=0: [(1)];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 3,(7), 15, 31, 63, 127, 255, 511, 1023, ...];
n=3: [1, 7, 35,(155), 651, 2667, 10795, 43435, 174251, ...];
n=4: [1, 15, 155, 1395,(11811), 97155, 788035, 6347715, ...];
n=5: [1, 31, 651, 11811, 200787,(3309747), 53743987, ...];
n=6: [1, 63, 2667, 97155, 3309747, 109221651,(3548836819), ...];
n=7: [1, 127, 10795, 788035, 53743987, 3548836819, 230674393235,(14877590196755), ...]; ...
the coefficients in parenthesis give the initial terms of this sequence;
an adjacent diagonal forms the Gaussian binomial coefficients [2*n,n] for q=2:
[1, 3, 35, 1395, 200787, 109221651, 230674393235, ...] = A006098.

G. C. Greubel, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A022166, A006098, A006125.

Table[QBinomial[2n-1, n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 12 2016 *)

{a(n)=polcoeff(prod(k=0,n-1,1/(1-2^k*x +x*O(x^n))),n)}
for(n=0,20,print1(a(n),", "))

a(n) = [x^n] Product_{k=0..n-1} 1/(1 - 2^k*x).

a(n) ~ c * 2^(n*(n-1)), where c = A065446. - Vaclav Kotesovec, Sep 22 2016

A306204 Decimal expansion of Product_{p>=3} (1+1/p) over the Mersenne primes.

Original entry on oeis.org

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

Offset: 1

Tomohiro Yamada, Jan 29 2019

This is equal to Product_{q>=1} (1-1/2^q)^(-1) over all q with 2^q - 1 a Mersenne prime.

			Decimal expansion of (4/3) * (8/7) * (32/31) * (128/127) * (8192/8191) * (131072/131071) * (524288/524287) * ... = 1.585558887...

Tomohiro Yamada, Table of n, a(n) for n = 1..99
Tomohiro Yamada, Unitary super perfect numbers, Mathematica Pannonica, Volume 19, No. 1, 2008, pp. 37-47, using this constant with only a rough upper bound (4/3)*exp(4/21) < 1.6131008.

Cf. A065446 (the corresponding product over all Mersenne numbers, prime or composite).
Cf. A173898 (the sum of reciprocals of the Mersenne primes).
Cf. A065442 (the sum of reciprocals of the Mersenne numbers, prime or composite).
Cf. A046528.

t=1.0;for(i=1,500,p=2^i-1;if(isprime(p),t=t*(p+1)/p))

Equals Sum_{n>=1} 1/A046528(n). - Amiram Eldar, Jan 06 2021

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)).

A356282 a(n) = Sum_{k=0..n} binomial(3n, n-k) p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 4, 23, 141, 888, 5675, 36602, 237563, 1548995, 10135554, 66504699, 437359454, 2881641263, 19016505326, 125664684700, 831400186740, 5506287269802, 36501297800013, 242167539749593, 1607851773270316, 10682384379036741, 71016046921543562, 472376627798814453

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000041, A188675, A356280, A356283.

Table[Sum[PartitionsP[k]*Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]

a(n) = sum(k=0, n, binomial(3*n, n-k)*numbpart(k)); \\ Michel Marcus, Aug 02 2022

a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} p(j)/2^j = A065446 = 3.4627466194550636115379573429...

A371747 Decimal expansion of Product_{k>=1} 1 / (1 - 1/4^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			1.4523536424495970158347130224852748733612...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000302, A065446, A100221, A370466, A371748, A371749, A371750.

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

Equals 1 / A100221.

A371750 Decimal expansion of Product_{k>=1} 1 / (1 - 1/5^k).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2024

			1.3152135557353452193080945851009617844...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A000351, A065446, A100222, A370466, A371747, A371751, A371752.

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

Equals 1 / A100222.

A006099 Gaussian binomial coefficient [ n, n/2 ] for q=2.

Original entry on oeis.org

1, 1, 3, 7, 35, 155, 1395, 11811, 200787, 3309747, 109221651, 3548836819, 230674393235, 14877590196755, 1919209135381395, 246614610741341843, 63379954960524853651, 16256896431763117598611, 8339787869494479328087443, 4274137206973266943778085267

Offset: 0

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Table of n, a(n) for n=0..50.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, q-Binomial Coefficient.

Cf. A065446.

Table[QBinomial[n,Floor[n/2],2],{n,0,20}] (* Harvey P. Dale, Sep 07 2013 *)

a(n) ~ c * 2^(n^2/4), where c = 1 / QPochhammer[1/2, 1/2] = A065446 = 3.46274661945506361153795734292443116454... if n is even, and c = 2^(-1/4) / QPochhammer[1/2, 1/2] = 2^(-1/4) * A065446 = 2.911811219231681420726836976930855961516... if n is odd. - Vaclav Kotesovec, Jun 22 2014

More terms from Harvey P. Dale, Sep 07 2013

A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

Original entry on oeis.org

1, 4, 24, 112, 544, 2368, 10624, 44800, 190976, 791552, 3282944, 13414400, 54829056, 222117888, 899383296, 3625123840, 14601027584, 58659700736, 235555782656, 944552017920, 3786334535680, 15166305468416, 60736264994816, 243129089261568, 973133053952000

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

Cf. A075900, A300579.

Cf. A023882, A077365, A179381, A300520.

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

a(n) ~ c * 4^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361...

A300583 Expansion of Product_{k>=1} 1 / (1 - 23^kx^k).

Original entry on oeis.org

1, 6, 54, 378, 2754, 17982, 121014, 765450, 4894506, 30429918, 189311094, 1160312850, 7113869226, 43228473822, 262556300286, 1587419581410, 9590551158474, 57795130268694, 348125978482686, 2093918636332530, 12590534397102930, 75647788993941174

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

Cf. A242587, A300582.

nmax = 25; CoefficientList[Series[Product[1/(1-2*3^k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 6^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361153...

A330863 Decimal expansion of Product_{k>=1} (1 + 1/(-2)^k).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			(1 - 1/2) * (1 + 1/2^2) * (1 - 1/2^3) * (1 + 1/2^4) * (1 - 1/2^5) * ... = 0.568698946265428505954976737...

Cf. A000593, A027637, A048651, A062510, A065446, A079555, A081845, A085011, A100221, A132020, A330862, A216206.

RealDigits[QPochhammer[-1, -1/2]/2, 10, 110] [[1]]
N[3/QPochhammer[-2, 1/4], 120] (* Vaclav Kotesovec, Apr 28 2020 *)

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

Equals Product_{k>=1} 1/(1 + 1/2^(2*k-1)).
Equals exp(Sum_{k>=1} A000593(k)/(k*(-2)^k)).
From Peter Bala, Dec 15 2020: (Start)
Constant C = (2/3) - (1/3)*Sum_{n >= 0} (-1)^n * 2^(n^2)/( Product_{k = 1..n+1} 4^k - 1 ).
C = Sum_{n >= 0} 1/( Product_{k = 1..n} (-2)^k - 1 ) = 1 - 1/3  - 1/9 + 1/81 + 1/1215 - - + + ... = Sum_{n >= 0} 1/A216206(n).
C = 1 + Sum_{n >= 0} (-1/2)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
3*C = 2 - Sum_{n >= 0} (1/4)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
9*C = 5 - Sum_{n >= 0} (-1/8)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
81*C = 46 + Sum_{n >= 0} (1/16)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
1215*C = 691 + Sum_{n >= 0} (-1/32)^(n+1)*Product_{k = 1..n} (1 + (-1/2)^k).
The sequence [1, 2, 5, 46, 691, ...] is the sequence of numerators of the  partial sums of the series Sum_{n >= 0} 1/A216206(n). (End)

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A218449 Gaussian binomial coefficient [2*n-1,n] for q=2, n>=0.

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A306204 Decimal expansion of Product_{p>=3} (1+1/p) over the Mersenne primes.

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

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A356282 a(n) = Sum_{k=0..n} binomial(3*n, n-k) * p(k), where p(k) is the partition function A000041.

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A371747 Decimal expansion of Product_{k>=1} 1 / (1 - 1/4^k).

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A371750 Decimal expansion of Product_{k>=1} 1 / (1 - 1/5^k).

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A006099 Gaussian binomial coefficient [ n, n/2 ] for q=2.

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A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

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A356282 a(n) = Sum_{k=0..n} binomial(3n, n-k) p(k), where p(k) is the partition function A000041.

A300583 Expansion of Product_{k>=1} 1 / (1 - 23^kx^k).