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A005329 a(n) = Product_{i=1..n} (2^i - 1). Also called 2-factorial numbers.
(Formerly M3085)
61
1, 1, 3, 21, 315, 9765, 615195, 78129765, 19923090075, 10180699028325, 10414855105976475, 21319208401933844325, 87302158405919092510875, 715091979502883286756577125, 11715351900195736886933003038875, 383876935713713710574133710574817125 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

A. Cuyt, V. B. Petersen, B. Verdonk, H. Waadeland, and W. B. Jones. Handbook of continued fractions for special functions. Springer, New York, 2008. (see 19.2.1)

M. 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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..50

E. Andresen, K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

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

FORMULA

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

EXAMPLE

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

MAPLE

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);

MATHEMATICA

a[0] = 1; a[n_] := a[n] = (2^n-1)*a[n-1]; a /@ Range[0, 14] (* Jean-Fran├žois 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 *)

PROG

(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

(PARI) 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

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

(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 */

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

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

CROSSREFS

Cf. A000225, A005321, A005329, A006125, A114604, A006088, A028362, A027871 (3-fac), A027872 (5-fac), A027873 (6-fac), A048651, A048652, A075271, A075272, A032085, A122746.

Sequence in context: A158888 A331583 A305532 * A134528 A332974 A118410

Adjacent sequences:  A005326 A005327 A005328 * A005330 A005331 A005332

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified September 23 22:32 EDT 2020. Contains 337315 sequences. (Running on oeis4.)