%I M3085 #158 Feb 19 2022 07:24:12

%S 1,1,3,21,315,9765,615195,78129765,19923090075,10180699028325,

%T 10414855105976475,21319208401933844325,87302158405919092510875,

%U 715091979502883286756577125,11715351900195736886933003038875,383876935713713710574133710574817125

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

%C 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

%C 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

%C Number of upper triangular n X n (0,1)-matrices with no zero rows. - _Vladeta Jovovic_, Mar 10 2008

%C 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

%C 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

%C 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

%C Also the number of dominating sets in the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017

%C 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

%C 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

%D 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)

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 358.

%D Mark Ronan, Lectures on Buildings (Perspectives in Mathematics; Vol. 7), Academic Press Inc., 1989.

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

%H T. D. Noe, <a href="/A005329/b005329.txt">Table of n, a(n) for n = 0..50</a>

%H E. Andresen and K. Kjeldsen, <a href="http://dx.doi.org/10.1016/0012-365X(76)90054-6">On certain subgraphs of a complete transitively directed graph</a>, Discrete Math., Vol. 14, No. 2 (1976), pp. 103-119.

%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.

%H Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.

%H Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, <a href="https://arxiv.org/abs/2012.13570">Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow</a>, arXiv:2012.13570 [math.CO], 2020.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DominatingSet.html">Dominating Set</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F a(n)/2^(n*(n+1)/2) -> c = 0.2887880950866024212788997219294585937270... (see A048651, A048652).

%F From _Paul D. Hanna_, Sep 17 2009: (Start)

%F G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n / (Product_{k=0..n} (1+2^k*x)).

%F Compare to: 1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/(Product_{k=1..n+1} (1+2^k*x)). (End)

%F 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

%F 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

%F a(n) = Product_{i=1..n} A000225(i). - _Michel Marcus_, Dec 27 2015

%F From _Peter Bala_, Nov 10 2017: (Start)

%F 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).

%F 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)

%F 0 = a(n)*(a(n+1) - a(n+2)) + 2*a(n+1)^2 for all n>=0. - _Michael Somos_, Feb 23 2019

%F From _Amiram Eldar_, Feb 19 2022: (Start)

%F Sum_{n>=0} 1/a(n) = A079555.

%F Sum_{n>=0} (-1)^n/a(n) = A048651. (End)

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

%p 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);

%t a[0] = 1; a[n_] := a[n] = (2^n-1)*a[n-1]; a /@ Range[0,14] (* _Jean-François Alcover_, Apr 22 2011 *)

%t FoldList[Times, 1, 2^Range[15] - 1] (* _Harvey P. Dale_, Dec 21 2011 *)

%t Table[QFactorial[n, 2], {n, 0, 14}] (* _Arkadiusz Wesolowski_, Oct 30 2012 *)

%t QFactorial[Range[0, 10], 2] (* _Eric W. Weisstein_, Jul 14 2017 *)

%t a[ n_] := If[ n < 0, 0, (-1)^n QPochhammer[ 2, 2, n]]; (* _Michael Somos_, Jan 28 2018 *)

%o (PARI) 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

%o (PARI) Dx(n,F)=local(D=F);for(i=1,n,D=deriv(D));D

%o 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

%o (PARI) {a(n) = if( n<0, 0, prod(k=1, n, 2^k - 1))}; /* _Michael Somos_, Jan 28 2018 */

%o (PARI) {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 */

%o (Magma) [1] cat [&*[ 2^k-1: k in [1..n] ]: n in [1..16]]; // _Vincenzo Librandi_, Dec 24 2015

%o (GAP) List([0..15],n->Product([1..n],i->2^i-1)); # _Muniru A Asiru_, May 18 2018

%Y Cf. A000225, A005321, A006125, A114604, A006088, A028362, A027871 (3-fac), A027872 (5-fac), A027873 (6-fac), A048651, A048652, A075271, A075272, A032085, A122746.

%Y Cf. A048651, A079555, A152476 (inverse binomial transform).

%Y Column q=2 of A069777.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

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