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

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.

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

[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

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.

Sequence in context: A008863 A133025 A118655 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.