A027871 -id:A027871 - cp's OEIS frontend

A005329 a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.

1, 1, 3, 21, 315, 9765, 615195, 78129765, 19923090075, 10180699028325, 10414855105976475, 21319208401933844325, 87302158405919092510875, 715091979502883286756577125, 11715351900195736886933003038875, 383876935713713710574133710574817125

Offset: 0

N. J. A. Sloane

Conjecture: this sequence is the inverse binomial transform of A075272 or, equivalently, the inverse binomial transform of the BinomialMean transform of A075271. - John W. Layman, Sep 12 2002
To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125. The probability of having won before n+1 tails is A114604 / A006125. - Joshua Zucker, Dec 14 2005
Number of upper triangular n X n (0,1)-matrices with no zero rows. - Vladeta Jovovic, Mar 10 2008
Equals the q-Fibonacci series for q = (-2), and the series prefaced with a 1: (1, 1, 1, 3, 21, ...) dot (1, -2, 4, -8, ...) if n is even, and (-1, 2, -4, 8, ...) if n is odd. For example, a(3) = 21 = (1, 1, 1, 3) dot (-1, 2, -4, 8) = (-1, 2, -4, 24) and a(4) = 315 = (1, 1, 1, 3, 21) dot (1, -2, 4, -8 16) = (1, -2, 4, -24, 336). - Gary W. Adamson, Apr 17 2009
Number of chambers in an A_n(K) building where K=GF(2) is the field of two elements. This is also the number of maximal flags in an n-dimensional vector space over a field of two elements. - Marcos Spreafico, Mar 22 2012
Given probability p = 1/2^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, A114604(n)/A006125(n+2) = 1-a(n)/A006125(n+1) is the probability that the outcome has occurred up to and including the n-th iteration. The limiting ratio is 1-A048651 ~ 0.7112119. These observations are a more formal and generalized statement of Joshua Zucker's Dec 14, 2005 comment. - Bob Selcoe, Mar 02 2016
Also the number of dominating sets in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Empirical: Letting Q denote the Hall-Littlewood Q basis of the symmetric functions over the field of fractions of the univariate polynomial ring in t over the field of rational numbers, and letting h denote the complete homogeneous basis, a(n) is equal to the absolute value of 2^A000292(n) times the coefficient of h_{1^(n*(n+1)/2)} in Q_{(n, n-1, ..., 1)} with t evaluated at 1/2. - John M. Campbell, Apr 30 2018
The series f(x) = Sum_{n>=0} x^(2^n-1)/a(n) satisfies f'(x) = f(x^2), f(0) = 1. - Lucas Larsen, Jan 05 2022

			G.f. = 1 + x + 3*x^2 + 21*x^3 + 315*x^4 + 9765*x^5 + 615195*x^6 + 78129765*x^7 + ...

Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones, Handbook of continued fractions for special functions, Springer, New York, 2008. (see 19.2.1)
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.
Mark Ronan, Lectures on Buildings (Perspectives in Mathematics; Vol. 7), Academic Press Inc., 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math., Vol. 14, No. 2 (1976), pp. 103-119.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Eric Weisstein's World of Mathematics, Dominating Set.
Eric Weisstein's World of Mathematics, q-Factorial.
Index entries for sequences related to factorial numbers.

Cf. A000225, A005321, A006125, A114604, A006088, A028362, A027871 (3-fac), A027872 (5-fac), A027873 (6-fac), A048651, A048652, A075271, A075272, A032085, A122746.
Cf. A048651, A079555, A152476 (inverse binomial transform).
Column q=2 of A069777.

List([0..15],n->Product([1..n],i->2^i-1)); # Muniru A Asiru, May 18 2018

[1] cat [&*[ 2^k-1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015

A005329 := proc(n) option remember; if n<=1 then 1 else (2^n-1)*procname(n-1); end if; end proc: seq(A005329(n), n=0..15);

a[0] = 1; a[n_] := a[n] = (2^n-1)*a[n-1]; a /@ Range[0,14] (* Jean-François Alcover, Apr 22 2011 *)
FoldList[Times, 1, 2^Range[15] - 1] (* Harvey P. Dale, Dec 21 2011 *)
Table[QFactorial[n, 2], {n, 0, 14}] (* Arkadiusz Wesolowski, Oct 30 2012 *)
QFactorial[Range[0, 10], 2] (* Eric W. Weisstein, Jul 14 2017 *)
a[ n_] := If[ n < 0, 0, (-1)^n QPochhammer[ 2, 2, n]]; (* Michael Somos, Jan 28 2018 *)

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, Sep 17 2009

Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D
a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+sum(k=1,n,x^k/k!*Dx(k,x*A+x*O(x^n) ))); polcoeff(A,n) \\ Paul D. Hanna, Apr 21 2012

{a(n) = if( n<0, 0, prod(k=1, n, 2^k - 1))}; /* Michael Somos, Jan 28 2018 */

{a(n) = if( n<0, 0, (-1)^n * sum(k=0, n+1, (-1)^k * 2^(k*(k+1)/2) * prod(j=1, k, (2^(n+1-j) - 1) / (2^j - 1))))}; /* Michael Somos, Jan 28 2018 */

a(n)/2^(n*(n+1)/2) -> c = 0.2887880950866024212788997219294585937270... (see A048651, A048652).
From Paul D. Hanna, Sep 17 2009: (Start)
G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n / (Product_{k=0..n} (1+2^k*x)).
Compare to: 1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/(Product_{k=1..n+1} (1+2^k*x)). (End)
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x). - Paul D. Hanna, Apr 21 2012
a(n) = 2^(binomial(n+1,2))*(1/2; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A000225(i). - Michel Marcus, Dec 27 2015
From Peter Bala, Nov 10 2017: (Start)
O.g.f. as a continued fraction of Stieltjes' type: A(x) = 1/(1 - x/(1 - 2*x/(1 - 6*x/(1 - 12*x/(1 - 28*x/(1 - 56*x/(1 - ... -(2^n - 2^floor(n/2))*x/(1 - ... )))))))) (follows from Heine's continued fraction for the ratio of two q-hypergeometric series at q = 2. See Cuyt et al. 19.2.1).
A(x) = 1/(1 + x - 2*x/(1 - (2 - 1)^2*x/(1 + x - 2^3*x/(1 - (2^2 - 1)^2*x/(1 + x - 2^5*x/(1 - (2^3 - 1)^2*x/(1 + x - 2^7*x/(1 - (2^4 - 1)^2*x/(1 + x - ... ))))))))). (End)
0 = a(n)*(a(n+1) - a(n+2)) + 2*a(n+1)^2 for all n>=0. - Michael Somos, Feb 23 2019
From Amiram Eldar, Feb 19 2022: (Start)
Sum_{n>=0} 1/a(n) = A079555.
Sum_{n>=0} (-1)^n/a(n) = A048651. (End)

Better definition from Leslie Ann Goldberg (leslie(AT)dcs.warwick.ac.uk), Dec 11 1999

A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 09 2004

Limit of the probability that a random N X N matrix, with entries chosen independently and uniformly from the field F_3, is nonsingular [Morrison (2006)]. - L. Edson Jeffery, Jan 22 2012

			0.56012607792794894496979224331414001437973633379836...

G. C. Greubel, Table of n, a(n) for n = 0..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.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A048651, A027871.

Cf. A000203, A100221, A100222, A132019, A132034, A132035, A132036, A132037, A132038, A258458.

N[(3^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[3]]^(1/3))/2^(1/3)]
N[QPochhammer[1/3,1/3]] (* G. C. Greubel, Nov 27 2015 *)

exp(-Sum_{k > 0} sigma_1(k)/k/3^k) = exp(-Sum_{k > 0} A000203(k)/k/3^k). - Hieronymus Fischer, Aug 07 2007
Product_{k >= 1} (1 - 1/3^k) = (1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
From Peter Bala, Jan 18 2021: (Start)
Constant C = (1 - 1/3)*Sum_{n >= 0} (-1/3)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*Sum_{n >= 0} (-1/9)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*(1 - 1/27)*Sum_{n >= 0} (-1/27)^n/Product_{k = 1..n} (3^k - 1), and so on. (End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(3)) * exp(log(3)/24 - Pi^2/(6*log(3))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027871(n). (End)

A053290 Number of nonsingular n X n matrices over GF(3).

Original entry on oeis.org

1, 2, 48, 11232, 24261120, 475566474240, 84129611558952960, 134068444202678083338240, 1923442429811445711790394572800, 248381049201184165590947520186915225600, 288678833735376059528974260112416365258106470400

Offset: 0

Stephen G Penrice, Mar 04 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..45
Jeffrey Overbey, William Traves, and Jerzy Wojdylo, On the Keyspace of the Hill Cipher, Cryptologia, Vol. 29, Iss. 1 (2005), pp. 59-72; author's copy.

Column k=3 of A316622.
Cf. A002884, A003787, A027871, A047656, A053291, A053292, A053293.
Cf. A100220, A219205.

[1] cat [&*[(3^n - 3^k): k in [0..n-1]]: n in [1..9]]; // Bruno Berselli, Jan 28 2013

Table[Product[3^n - 3^k, {k, 0, n - 1}], {n, 0, 10}] (* Geoffrey Critzer, Jan 26 2013; edited by Vincenzo Librandi, Jan 28 2013 *)

for(n=0,10, print1(prod(k=0,n-1, 3^n - 3^k), ", ")) \\ G. C. Greubel, May 31 2018

a(n) = Product_{k=0..n-1}(3^n-3^k). - corrected by Michel Marcus, Sep 18 2015
a(n) = A047656(n)*A027871(n). - Bruno Berselli, Jan 30 2013
From Amiram Eldar, Jul 06 2025: (Start)
a(n) = Product_{k=1..n} A219205(k).
a(n) ~ c * 3^(n^2), where c = A100220. (End)

More terms from Vladeta Jovovic, Mar 16 2000

A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 130, 40, 1, 1, 121, 1210, 1210, 121, 1, 1, 364, 11011, 33880, 11011, 364, 1, 1, 1093, 99463, 925771, 925771, 99463, 1093, 1, 1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1

Offset: 0

N. J. A. Sloane

The coefficients of the matrix inverse are apparently given by T^(-1)(n,k) = (-1)^n*A157783(n,k). - R. J. Mathar, Mar 12 2013

			Triangle begins:
  1;
  1,    1;
  1,    4,      1;
  1,   13,     13,        1;
  1,   40,    130,       40,        1;
  1,  121,   1210,     1210,      121,        1;
  1,  364,  11011,    33880,    11011,      364,      1;
  1, 1093,  99463,   925771,   925771,    99463,   1093,    1;
  1, 3280, 896260, 25095280, 75913222, 25095280, 896260, 3280, 1;

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Rows n = 0..50 of triangle, flattened
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Index entries for sequences related to Gaussian binomial coefficients

Columns k=0..3 give A000012, A003462, A006100, A006101.

Cf. A006117 (row sums).

A022167 := proc(n,m)
        A027871(n)/A027871(n-m)/A027871(m) ;
end proc:
seq(seq(A022167(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

p[n_] := Product[3^k-1, {k, 1, n}]; t[n_, m_] := p[n]/(p[n-m]*p[m]); Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 10 2014, after R. J. Mathar *)
Table[QBinomial[n, k, 3], {n, 0, 10}, {k, 0, n}] // Flatten
(* or, after Vladimir Kruchinin, using S for qStirling2: *)
S[n_, k_, q_] /; 1 <= k <= n := S[n-1, k-1, q] + Sum[q^j, {j, 0, k-1}]* S[n-1, k, q]; S[n_, 0, ] := KroneckerDelta[n, 0]; S[0, k, ] := KroneckerDelta[0, k]; S[, , ] = 0;
T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n-j, n-k, q]*(q-1)^(k-j) /. q -> 3, {j, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020 *)

T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
T(n,k) = Sum_{j=0..k} C(n,j)*qStirling2(n-j,n-k,3)*(2)^(k-j),j,0,k), n >= k, where qStirling2(n,k,3) is triangle A333143. - Vladimir Kruchinin, Mar 07 2020
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 1. - Seiichi Manyama, May 09 2025

A027637 a(n) = Product_{i=1..n} (4^i - 1).

Original entry on oeis.org

1, 3, 45, 2835, 722925, 739552275, 3028466566125, 49615367752825875, 3251543125681443718125, 852369269595510700600441875, 893773106866112632882108339078125, 3748755223447856814435325652920396921875

Offset: 0

N. J. A. Sloane

The q-analog of double factorials (A000165) evaluated at q=2. - Michael Somos, Sep 12 2014
3^n*5^(floor(n/2))|a(n) for n>=1. - G. C. Greubel, Nov 21 2015
Given probability p = 1/4^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1-a(n)/A053763(n+1) is the probability that the outcome has occurred up to and including the n-th iteration. The limiting ratio is 1-A100221 ~ 0.3114625. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A000165.
Sequences of the form q-Pochhammer(n, q, q): A005329 (q=2), A027871 (q=3), this sequence (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).
Cf. A053763, A100221.

[1] cat [&*[4^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027637 := proc(n)
    mul( 4^i-1,i=1..n) ;
end proc:
seq(A027637(n),n=0..8) ;

A027637 = Table[Product[4^i - 1, {i, n}], {n, 0, 9}] (* Alonso del Arte, Nov 14 2011 *)
a[ n_] := If[ n < 0, 0, Product[ (q^(2 k) - 1) / (q - 1), {k, n}] /. q -> 2]; (* Michael Somos, Sep 12 2014 *)
Abs@QPochhammer[4, 4, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = prod(i=1, n, 4^i-1); \\ Michel Marcus, Nov 21 2015

from sage.combinat.q_analogues import q_pochhammer
def A027637(n): return (-1)^n*q_pochhammer(n, 4, 4)
[A027637(n) for n in (0..15)] # G. C. Greubel, Aug 04 2022

a(n) ~ c * 2^(n*(n+1)), where c = Product_{k>=1} (1-1/4^k) = A100221 = 0.688537537120339715456514357293508184675549819378... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 4^(binomial(n+1,2))*(1/4;1/4){n} = (4; 4){n}, where (a;q){n} is the q-Pochhammer symbol. - _G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 4^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 4^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A100221. - Amiram Eldar, May 07 2023

A027878 a(n) = Product_{i=1..n} (10^i - 1).

Original entry on oeis.org

1, 9, 891, 890109, 8900199891, 890011088900109, 890010198889020099891, 8900101098880002109889900109, 890010100987899112108987901010099891, 890010100097889011121088788901111989989900109

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027879 (q=11), A027880 (q=12).

Cf. A132038, A132326.

[1] cat [&*[10^k-1: k in [1..n]]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Table[Product[10^i-1,{i,n}],{n,0,10}] (* Harvey P. Dale, Aug 15 2011 *)
Abs@QPochhammer[10, 10, Range[0, 30]] (* G. C. Greubel, Nov 24 2015 *)

a(n) = prod(k=1, n, 10^k - 1) \\ Altug Alkan, Nov 25 2015

a(n) ~ c * 10^(n*(n+1)/2), where c = Product_{k>=1} (1-1/10^k) = A132038 = 0.890010099998999000000100009999999989999900000000... . - Vaclav Kotesovec, Nov 21 2015
3^n*(11)^(floor(n/2)) divides a(n) for n>=0. - G. C. Greubel, Nov 24 2015
Equals 10^(binomial(n+1,2))*(1/10;1/10){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f.: Sum_{n>=0} 10^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 10^k*x). - Ilya Gutkovskiy, May 22 2017
From Amiram Eldar, May 07 2023: (Start)
Sum_{n>=0} 1/a(n) = A132326.
Sum_{n>=0} (-1)^n/a(n) = A132038. (End)

A027872 a(n) = Product_{i=1..n} (5^i - 1).

Original entry on oeis.org

1, 4, 96, 11904, 7428096, 23205371904, 362560730628096, 28324694519589371904, 11064305472020078810628096, 21609960560733744406929189371904, 211034749490954911990173458030810628096

Offset: 0

N. J. A. Sloane

nonn

Given probability p = 1/5^n that an outcome will occur at the n-th stage of an infinite process, then starting at n=1, 1 - a(n)/A109345(n+1) is the probability that the outcome has occurred at or before the n-th iteration. The limiting ratio is 1-A100222 ~ 0.2396672. - Bob Selcoe, Mar 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027873 (q=6), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A109345, A100222.

[1] cat [&*[ 5^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

A027872 := proc(n)
        mul( 5^i-1, i=1..n) ;
end proc: # R. J. Mathar, Mar 12 2013

Table[Product[(5^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[5, 5, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Join[{1},FoldList[Times,5^Range[10]-1]] (* Harvey P. Dale, Dec 28 2021 *)

a(n) = prod(i=1, n, 5^i-1); \\ Michel Marcus, Nov 21 2015

4^n|a(n) for n >= 1. - G. C. Greubel, Nov 21 2015
a(n) ~ c * 5^(n*(n+1)/2), where c = Product_{k>=1} (1-1/5^k) = A100222 . - Vaclav Kotesovec, Nov 21 2015
a(n) = 5^(binomial(n+1,2))*(1/5; 1/5){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
a(n) = Product_{i=1..n} A024049(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 5^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 5^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A100222. - Amiram Eldar, May 07 2023

A027873 a(n) = Product_{i=1..n} (6^i - 1).

Original entry on oeis.org

1, 5, 175, 37625, 48724375, 378832015625, 17674407688984375, 4947685316415841015625, 8310206472731792807458984375, 83747726219216824716765369541015625

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027875 (q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A132034.

[1] cat [&*[ 6^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Table[Product[(6^k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 17 2015 *)
Abs@QPochhammer[6, 6, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
FoldList[Times,Join[{1},6^Range[10]-1]] (* Harvey P. Dale, Oct 13 2017 *)

a(n) = prod(i=1, n, 6^i-1); \\ Michel Marcus, Nov 21 2015

5^n|a(n) for n>=0. - G. C. Greubel, Nov 20 2015
a(n) ~ c * 6^(n*(n+1)/2), where c = Product_{k>=1} (1-1/6^k) = A132034 = 0.805687728162164940923750215496298968917997628693... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 6^(binomial(n+1,2))*(1/6;1/6){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024062(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 6^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 6^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132034. - Amiram Eldar, May 07 2023

A027875 a(n) = Product_{i=1..n} (7^i - 1).

Original entry on oeis.org

1, 6, 288, 98496, 236390400, 3972777062400, 467389275837235200, 384914699001548351078400, 2218956256804125934296760320000, 89542886518308517126993353029713920000

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A132035.

[1] cat [&*[ 7^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

Abs@QPochhammer[7, 7, Range[0, 10]] (* Vladimir Reshetnikov, Nov 20 2015 *)
Table[Product[7^k-1,{k,n}],{n,0,10}] (* Harvey P. Dale, Jul 28 2022 *)

a(n) = prod(i=1, n, 7^i-1); \\ Michel Marcus, Nov 21 2015

2*(10)^(2m)|a(n) where 4*m <= n <= 4*m+3, for m >= 1. - G. C. Greubel, Nov 20 2015
a(n) ~ c * 7^(n*(n+1)/2), where c = Product_{k>=1} (1-1/7^k) = A132035 = 0.836795407089037871026729798146136241352436435876... . - Vaclav Kotesovec, Nov 21 2015
a(n) = 7^(binomial(n+1,2))*(1/7;1/7){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
a(n) = Product_{i=1..n} A024075(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 7^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 7^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132035. - Amiram Eldar, May 07 2023

A027876 a(n) = Product_{i=1..n} (8^i - 1).

Original entry on oeis.org

1, 7, 441, 225351, 922812345, 30237792108615, 7926625536728661945, 16623330670976050126618695, 278893192683059452825059069034425, 37432410397693271164043156886536608251975

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6), A027875 (q=7), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880 (q=12).

Cf. A132036.

[1] cat [&*[ 8^k-1: k in [1..n] ]: n in [1..11]]; // Vincenzo Librandi, Dec 24 2015

seq(mul(8^i-1,i=1..n), n=0..20); # Robert Israel, Dec 24 2015

FoldList[Times,1,8^Range[10]-1] (* Harvey P. Dale, Dec 23 2011 *)

a(n)=prod(i=1,n,8^i-1) \\ Charles R Greathouse IV, Nov 22 2015

a(n) ~ c * 8^(n*(n+1)/2), where c = Product_{k>=1} (1-1/8^k) = A132036 = 0.859405994400702866200758580064418894909484979588... . - Vaclav Kotesovec, Nov 21 2015
7^n | a(n). - G. C. Greubel, Nov 21 2015
It appears that 7^m | a(n) iff 7^m | (7n)!. - Robert Israel, Dec 24 2015
a(n) = 8^(binomial(n+1,2))*(1/8;1/8){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 24 2015
G.f. g(x) satisfies (1+x) g(x) = 1 + 8 x g(8x). - Robert Israel, Dec 24 2015
a(n) = Product_{i=1..n} A024088(i). - Michel Marcus, Dec 27 2015
G.f.: Sum_{n>=0} 8^(n*(n+1)/2)*x^n / Product_{k=0..n} (1 + 8^k*x). - Ilya Gutkovskiy, May 22 2017
Sum_{n>=0} (-1)^n/a(n) = A132036. - Amiram Eldar, May 07 2023

cp's OEIS Frontend

A005329 a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.

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A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).

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A053290 Number of nonsingular n X n matrices over GF(3).

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A022167 Triangle of Gaussian binomial coefficients [ n,k ] for q = 3.

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A027637 a(n) = Product_{i=1..n} (4^i - 1).

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A027878 a(n) = Product_{i=1..n} (10^i - 1).

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