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Search: id:A156554
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| A156554 |
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The number of integer sequences of length d = 2n+1 such that the sum of the terms is 0 and the sum of the absolute values of the terms is d-1. |
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24
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| 1, 6, 110, 2562, 66222, 1815506, 51697802, 1511679210, 45076309166, 1364497268946, 41800229045610, 1292986222651646, 40317756506959050, 1265712901796074842, 39965073938276694002, 1268208750951634765562
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Let b(n) = S(d,n) be the coordination sequence of the lattice A_d. Then this sequence is a(n) = S(2n,n). See Conway-Sloane. The sequence is defined by Couveignes et al.
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LINKS
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R. H. Hardin, Table of n, a(n) for n = 0..26
Index entries for sequences related to linear recurrences with constant coefficients
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).
J.-M. Couveignes, T. Ezome and R. Lercier. Elliptic periods and primality proving, (2008) (pdf.
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FORMULA
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a(n) = S(2n,n) where S(d,n) = Sum(k=0..d,C(d,k)^2*C(n-k+d-1)) from formula (22) in Conway-Sloane.
G.f. is: x*(1-x)*(1+x)/(x^4-19*x^3+41*x^2-19*x+1) and a(n)=(1/Sqrt[205])*((-(19 - Sqrt[205] - Sqrt[550 - 38*Sqrt[205]])^n - (19 - Sqrt[205] + Sqrt[550 - 38*Sqrt[205]])^n + (19 + Sqrt[205] - Sqrt[550 + 38*Sqrt[205]])^n + (19 + Sqrt[205] + Sqrt[550 + 38*Sqrt[205]])^n)/4^n). - Peter Pein, Feb 09 2009
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EXAMPLE
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For n = 1 the a(n) = 6 sequences are (1,-1,0),(-1,1,0),(1,0,-1),(-1,0,1),(0,1,-1),(0,-1,1)
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MAPLE
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S:=proc(d, n) add(binomial(d, k)^2*binomial(n-k+d-1, d-1), k=0..d); end proc; a:=n->S(2*n, n);
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MATHEMATICA
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Table[ Binomial[-1 + 3 n, -1 + 2 n] HypergeometricPFQ[{-2 n, -2 n, -n}, {1, 1 - 3 n}, 1], {n, 0, 10}] - Eric Weisstein, Feb 10 2009
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CROSSREFS
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a(n)=A103881(2n, n)
Sequence in context: A041149 A119814 A050884 this_sequence A112499 A024273 A024274
Adjacent sequences: A156551 A156552 A156553 this_sequence A156555 A156556 A156557
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KEYWORD
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easy,nice,nonn
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AUTHOR
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W. Edwin Clark (eclark(AT)math.usf.edu), Feb 09 2009
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