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
Examples
G.f. = 1 + x + 3*x^2 + 21*x^3 + 315*x^4 + 9765*x^5 + 615195*x^6 + 78129765*x^7 + ...
References
- 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).
Links
- 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.
Crossrefs
Programs
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GAP
List([0..15],n->Product([1..n],i->2^i-1)); # Muniru A Asiru, May 18 2018
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Magma
[1] cat [&*[ 2^k-1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015
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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);
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Mathematica
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 *)
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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
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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
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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 */
Formula
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)
Extensions
Better definition from Leslie Ann Goldberg (leslie(AT)dcs.warwick.ac.uk), Dec 11 1999
Comments