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COMMENTS
| 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, 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, Mar 18 2005
Binomial transform of (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...); inverse binomial transform of A007051 . - Philippe DELEHAM, 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, Sep 12 2008]
Prepend A089067 with a 1, getting (1, 1, 3, 5, 13, 23, 51,...) as polcoeff A(x); then (1, 1, 2, 4, 8, 16,...) = A(x)/A(x^2). [Gary W. Adamson, Feb 18 2010]
a(n) = A173921(A000079(n)). [From Reinhard Zumkeller, Mar 04 2010]
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the corner squares these vectors lead to the companion sequence A094373. [Johannes W. Meijer, Aug 15 2010]
a(n) = SUM(A093873(k)/A093875(k): 2^n<=k<2^(n+1)), sums of rows of the full tree of Kepler's harmonic fractions. [From Reinhard Zumkeller, Oct 17 2010]
From Paul Curtz, Jul 20 2011. (Start)
Array T(m,n)=2*T(m,n-1) + T(m-1,n)
1, 1, 2, 4, 8, 16, =a(n)
1, 3, 8, 20, 48, 112, =A001792,
1, 5, 18, 56, 160, 432, =A001793
1, 7, 32, 120, 400, 1232, =A001794,
1, 9, 50, 220, 840, 2912, =A006974,followed with A006975,A006976,
gives nonzero coefficients of Chebyshev polynomials of first kind A039991=
1,
1, 0,
2, 0, -1,
4, 0, -3, 0,
8, 0, -8, 0, 1.
T(m,n) third vertical: 2*n^2, n positive (A001105).
Fourth vertical appears in Janet table even rows,last vertical (A168342 array, A138509,rank 3,13,=A166911)). (End)
A131577(n) and differences are
0, 1, 2, 4, 8, 16,
1, 1, 2, 4, 8, 16, =a(n),
0, 1, 2, 4, 8, 16,
1, 1, 2, 4, 8, 16.
Number of 2-color necklaces of length 2n equal to their complemented reversal. For length 2n+1, the number is 0. [David W. Wilson, Jan 01 2012]
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FORMULA
| a(0)=1, a(n)=2^(n-1).
G.f.: (1-x)/(1-2*x).
E.g.f.: cosh(z)*exp(z)=(exp(2z)+1)/2.
a(0) = 1 and for n>0, a(n) = sum of all previous terms.
a(n)=Sum{k=0..n, binomial(n, 2*k)}. - Paul Barry, Feb 25 2003
a(n)=Sum{k=0..n, binomial(n, k)*(1+(-1)^k)/2 } - Paul Barry, May 27 2003
a(n)=floor((1+2^n)/2) - Toby Bartels (toby+sloane(AT)math.ucr.edu), Aug 27 2003
G.f.: sum(i>=0, x^i/(1-x)^i) - Jon Perry, Jul 10 2004
a(n)=sum{k=0..n, (-1)^(n-k)*binomial(k+1, n-k)*binomial(2*k, k)} - Paul Barry, Mar 18 2005
a(n) = Sum_{k, 0<=k<=[n/2]} A055830(n-k,k) . - Philippe DELEHAM, Oct 22 2006
a(n) = Sum_{k, 0<=k<=n} A098158(n,k) . - Philippe DELEHAM, Dec 04 2006
G.f.: 1/(1-(x+x^2+x^3+...)) [From Geoffrey Critzer, Aug 30 2008]
a(n)=A000079(n) - A131577(n) .
E.g.f.: (exp(2*x)+1)/2 = (G(0) + 1)/2 ; G(k) = 1 + 2*x/(2*k+1 - x*(2*k+1)/(x + (k+1)/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 03 2011
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