0,3

Apart from initial term, same as A000079 (powers of 2).

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.

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.

a(n+2)= number of distinct Boolean functions of n variables under action of symmetric group.

Also the number of unlabeled (1+2)-free posets. - Detlef Pauly, May 25 2003

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

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

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

Also, number of rationals in [0, 1) whose binary expansions terminate after n bits. - Brad Chalfan (brad(AT)chalfan.net), May 29 2006

Equals row sums of triangle A144157 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, 383-406, 1985, see pp. 392-393.

Index entries for sequences related to linear recurrences with constant coefficients

S. Heubach and T. Mansour, Counting rises, levels and drops in compositions

Index entries for sequences related to Boolean functions

Index entries for related partition-counting sequences

a(n) = sum_i[a(i)] with a(0) = 1.

a(n)=Sum{k=0..n, binomial(n, 2k)}. - Paul Barry (pbarry(AT)wit.ie), Feb 25 2003

a(n)=Sum{k=0..n, binomial(n, k)(1+(-1)^k)/2 } - Paul Barry (pbarry(AT)wit.ie), May 27 2003

G.f.: (1-x)/(1-2x). E.g.f.: cosh(z)*exp(z). a(0)=1, a(n)=2^(n-1).

a(n)=floor((1+2^n)/2) - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003

G.f.: sum(i=0, oo, x^i/(1-x)^i) - Jon Perry (perry(AT)globalnet.co.uk), Jul 10 2004

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

a(0) = 1; for n>0, a(n) = sum of all previous terms.

a(n)=Sum_{k, 0<=k<=[n/2]}A055830(n-k,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006

ceil(binomial(n+0,0)*2^(n-1)). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 01 2006

a(n) = Sum_{k, 0<=k<=n}A098158(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2006

G.f.: 1/1-(x+x^2+x^3+...) [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Aug 30 2008]

[seq (ceil(binomial(n+0, 0)*2^(n-1)), n=0..32)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 01 2006

a:=n->mul(2, k=0..n): seq(a(n), n=-2..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 22 2008

with(finance):seq(ceil(futurevalue(4, 1, n)), n=-3..29); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 24 2009]

f[s_] := Append[s, Ceiling[Plus @@ s]]; Nest[f, {1}, 32] (* or *)

CoefficientList[ Series[(1 - x)/(1 - 2x), {x, 0, 32}], x] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jul 07 2006)

(PARI) a(n)=if(n<1, n==0, 2^(n-1))

Cf. A051486.

Cf. A000670, A051486.

Row sums of triangle A100257.

Cf. A082140, A082141, A082138, A082139, A080951, A080929, A057711.

A144157 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 12 2008]

Sequence in context: A118655 A155559 A166687 this_sequence A034008 A123344 A131577

Adjacent sequences: A011779 A011780 A011781 this_sequence A011783 A011784 A011785

nonn,nice,easy

killough(AT)wagner.convex.com (Lee D. Killough). Additional comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 14, 2001.

Corrected typo . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 25 2008