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A093387
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2^(n-1)-binomial(n, floor(n/2)).
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8
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0, 0, 1, 2, 6, 12, 29, 58, 130, 260, 562, 1124, 2380, 4760, 9949, 19898, 41226, 82452, 169766, 339532, 695860, 1391720, 2842226, 5684452, 11576916, 23153832, 47050564, 94101128, 190876696, 381753392, 773201629, 1546403258, 3128164186, 6256328372, 12642301534
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OFFSET
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1,4
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COMMENTS
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Suppose n >= 3. Let e_1,...,e_n be n unit-vectors which generate Euclidean space R_n and let l_n = {x= sum a_i e_i | a_1 >= a_2 >= ... >= a_n >= 0 }. Consider the hypercube H_n with vertices h_1,...,h_{2^n} = {epsilon_1 e_1+...+ epsilon_n e_n}.
For each element x in l_n we build 2^n "statements" by taking the inner product of x with h_i. We call a statement true if (x,h_i)>0 and false if (x,h_i)<0. Two vectors x and y are indistinguishable if all statements produced by x and y are equal.
For each set of indistinguishable vectors we chose one vector, which is called the representative. The sequence gives the number of representatives.
Hankel transform is A127365. - Paul Barry, Jan 11 2007
Alternative description from Emeric Deutsch, May 30 2011: (Start)
Number of up-steps starting at level 0 in all dispersed Dyck paths of length n-1 (i.e. in Motzkin paths of length n-1 with no (1,0)-steps at positive heights). (End)
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LINKS
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Table of n, a(n) for n=1..35.
Matthijs Coster, Sequences
Matthijs Coster, Statements and Representatives, 2004.
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FORMULA
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a(n) = A000079(n-1)-A001405(n).
a(n+1)=sum{k=2..n, C(n,floor((n-k)/2)}; - Paul Barry, Jan 11 2007
a(2n) = 2*a(2n-1). - Emeric Deutsch, May 30 2011
a(n+1) = sum_{k>=0} k*A191310(n,k). - Emeric Deutsch, May 30 2011
G.f.: (1-sqrt(1-4*z^2))^2/(4*z*(1-2*z)). - Emeric Deutsch, May 30 2011
Conjecture: (n+1)*a(n) +2*(-n-1)*a(n-1) +4*(-n+2)*a(n-2) +8*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
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EXAMPLE
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a(5)=6 because, denoting U=(1,1), D=(1,-1), H=(1,0), in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+1+1+1+2+1=6 U steps starting at level 0. - Emeric Deutsch, May 30 2011
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CROSSREFS
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Sequence in context: A183467 A057582 A094779 * A195166 A181826 A143176
Adjacent sequences: A093384 A093385 A093386 * A093388 A093389 A093390
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KEYWORD
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nonn
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AUTHOR
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Matthijs Coster, Apr 29 2004
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EXTENSIONS
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Offset corrected, R. J. Mathar, Jun 04 2011
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STATUS
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approved
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