%I A005329 M3085
%S A005329 1,1,3,21,315,9765,615195,78129765,19923090075,10180699028325,
%T A005329 10414855105976475,21319208401933844325,87302158405919092510875,
%U A005329 715091979502883286756577125,11715351900195736886933003038875
%N A005329 Product(2^i - 1), i=1..n. Also called 2-factorial numbers.
%C A005329 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 (layman(AT)math.vt.edu), Sep 
               12 2002
%C A005329 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 
               (joshua.zucker(AT)stanfordalumni.org), Dec 14 2005
%C A005329 Number of upper triangular n X n (0,1)-matrices with no zero rows. - 
               Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 10 2008
%C A005329 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009: 
               (Start)
%C A005329 Equals the q-Fibonacci series for q = (-2), and the series prefaced with 
               a 1:
%C A005329 (1, 1, 1, 3, 21,...) dot (1, -2, 4, -8,...) if n is even, and (-1, 2, 
               -4, 8,...) if n is odd.
%C A005329 Examples: a(3) = 21 = (1, 1, 1, 3) dot (-1, 2, -4, 8) = (-1, 2, -4, 24).
%C A005329 a(4) = 315 = (1, 1, 1, 3, 21) dot (1, -2, 4, -8 16) = (1, -2, 4, -24, 
               336). (End)
%D A005329 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005329 Andresen, E.; Kjeldsen, K.; On certain subgraphs of a complete transitively 
               directed graph. Discrete Math. 14 (1976), no. 2, 103-119.
%D A005329 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A005329 T. D. Noe, <a href="b005329.txt">Table of n, a(n) for n=0..50</a>
%H A005329 <a href="Sindx_Fa.html#factorial">Index entries for sequences related 
               to factorial numbers</a>
%F A005329 Better description from Leslie Ann Goldberg (leslie(AT)dcs.warwick.ac.uk).
%F A005329 a(n)/2^(n*(n+1)/2) -> c = 0.2887880950866024212788997219294585937270... 
               (see A048651, A048652).
%F A005329 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Sep 17 2009: 
               (Start)
%F A005329 G.f.: Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=0..n} (1+2^k*x)].
%F A005329 contrast with:
%F A005329 1 = Sum_{n>=0} 2^(n*(n+1)/2) * x^n/[Product_{k=1..n+1} (1+2^k*x)]. (End)
%p A005329 A005329 := proc(n) option remember; if n=1 then 1 else (2^n-1)*A005329(n-1); 
               fi; end;
%p A005329 restart:with (combinat):a:=n->mul(stirling2(j,2), j=2..n): seq(a(n), 
               n=1..19);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 
               01 2009]
%p A005329 restart:a:= proc(n) option remember; if n=0 then 1 else add(binomial 
               (n,j)*a(n-1), j=0..n-1) fi end: seq (a(n), n=0..14);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 29 2009]
%o A005329 (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)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 17 2009]
%Y A005329 Cf. A006125, A114604.
%Y A005329 Cf. A005321.
%Y A005329 Cf. A006088, A028362. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 
               17 2009]
%Y A005329 Sequence in context: A000681 A055555 A158888 this_sequence A134528 A118410 
               A125054
%Y A005329 Adjacent sequences: A005326 A005327 A005328 this_sequence A005330 A005331 
               A005332
%K A005329 nonn,easy,nice
%O A005329 0,3
%A A005329 N. J. A. Sloane (njas(AT)research.att.com).
%E A005329 More terms from Olivier Gerard 8/97.