Search: id:A011782
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%I A011782
%S A011782 1,1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A011782 131072,262144,524288,1048576,2097152,4194304,8388608,16777216,33554432,
%U A011782 67108864,134217728,268435456,536870912,1073741824,2147483648
%N A011782 Expansion of (1-x)/(1-2x) in powers of x.
%C A011782 Apart from initial term, same as A000079 (powers of 2).
%C A011782 Number of ways of putting n unlabeled items into (any number of) labeled 
               boxes where every box contains at least one item. Also "unimodal 
               permutations of n items", i.e. those which rise then fall. (E.g. 
               for three items: ABC, ACB, BCA and CBA are unimodal) - Henry Bottomley 
               (se16(AT)btinternet.com), Jan 17 2001.
%C A011782 Number of permutations in S_n avoiding the patterns 213 and 312. - Tuwani 
               Albert Tshifhumulo (tat(AT)univen.ac.za), Apr 20 2001. More generally 
               (see Simion and Schmidt), the number of permutations in S_n avoiding 
               (i) the 123 and 132 patterns; (ii) the 123 and 213 patterns; (iii) 
               the 132 and 213 patterns; (iv) the 132 and 231 patterns; (v) the 
               132 and 312 patterns; (vi) the 213 and 231 patterns; (vii) the 213 
               and 312 patterns; (viii) the 231 and 312 patterns; (ix) the 231 and 
               321 patterns; (x) the 312 and 321 patterns.
%C A011782 a(n+2)= number of distinct Boolean functions of n variables under action 
               of symmetric group.
%C A011782 Also the number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 
               2003
%C A011782 Also the number of compositions (ordered partitions) of n, so that (for 
               example) 3 = 2 + 1 and 3 = 1 + 2 are counted separately (but see 
               A000079). - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
%C A011782 Image of the central binomial coefficients A000984 under the Riordan 
               array ((1-x),x(1-x)). - Paul Barry (pbarry(AT)wit.ie), Mar 18 2005
%C A011782 Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial 
               transform of A007051 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jul 04 2005
%C A011782 Also, number of rationals in [0, 1) whose binary expansions terminate 
               after n bits. - Brad Chalfan (brad(AT)chalfan.net), May 29 2006
%C A011782 Equals row sums of triangle A144157 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Sep 12 2008]
%D A011782 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A011782 R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 
               6, 383-406, 1985, see pp. 392-393.
%H A011782 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A011782 S. Heubach and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0310197">
               Counting rises, levels and drops in compositions</a>
%H A011782 <a href="Sindx_Bo.html#Boolean">Index entries for sequences related to 
               Boolean functions</a>
%H A011782 <a href="Sindx_Par.html#partN">Index entries for related partition-counting 
               sequences</a>
%F A011782 a(n) = sum_i[a(i)] with a(0) = 1.
%F A011782 a(n)=Sum{k=0..n, binomial(n, 2k)}. - Paul Barry (pbarry(AT)wit.ie), Feb 
               25 2003
%F A011782 a(n)=Sum{k=0..n, binomial(n, k)(1+(-1)^k)/2 } - Paul Barry (pbarry(AT)wit.ie), 
               May 27 2003
%F A011782 G.f.: (1-x)/(1-2x). E.g.f.: cosh(z)*exp(z). a(0)=1, a(n)=2^(n-1).
%F A011782 a(n)=floor((1+2^n)/2) - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 
               27 2003
%F A011782 G.f.: sum(i=0, oo, x^i/(1-x)^i) - Jon Perry (perry(AT)globalnet.co.uk), 
               Jul 10 2004
%F A011782 a(n)=sum{k=0..n, (-1)^(n-k)binomial(k+1, n-k)binomial(2k, k)} - Paul 
               Barry (pbarry(AT)wit.ie), Mar 18 2005
%F A011782 a(0) = 1; for n>0, a(n) = sum of all previous terms.
%F A011782 a(n)=Sum_{k, 0<=k<=[n/2]}A055830(n-k,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 22 2006
%F A011782 ceil(binomial(n+0,0)*2^(n-1)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 01 2006
%F A011782 a(n) = Sum_{k, 0<=k<=n}A098158(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 04 2006
%F A011782 G.f.: 1/1-(x+x^2+x^3+...) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Aug 30 2008]
%p A011782 [seq (ceil(binomial(n+0,0)*2^(n-1)),n=0..32)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 01 2006
%p A011782 a:=n->mul(2, k=0..n): seq(a(n), n=-2..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 22 2008
%p A011782 with(finance):seq(ceil(futurevalue(4,1,n)), n=-3..29);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2009]
%t A011782 f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* or *)
%t A011782 CoefficientList[ Series[(1 - x)/(1 - 2x), {x, 0, 32}], x] (from Robert 
               G. Wilson v (rgwv(at)rgwv.com), Jul 07 2006)
%o A011782 (PARI) a(n)=if(n<1,n==0,2^(n-1))
%Y A011782 Cf. A051486.
%Y A011782 Cf. A000670, A051486.
%Y A011782 Row sums of triangle A100257.
%Y A011782 Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.
%Y A011782 A144157 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]
%Y A011782 Sequence in context: A118655 A155559 A166687 this_sequence A034008 A123344 
               A131577
%Y A011782 Adjacent sequences: A011779 A011780 A011781 this_sequence A011783 A011784 
               A011785
%K A011782 nonn,nice,easy
%O A011782 0,3
%A A011782 killough(AT)wagner.convex.com (Lee D. Killough). Additional comments 
               from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 14, 2001.
%E A011782 Corrected typo . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 25 
               2008

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