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Search: id:A158380
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| A158380 |
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Number of solutions to +-1+-3+-6..+-n(n+1)/2=0. |
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+0
2
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| 0, 0, 0, 2, 0, 2, 2, 4, 0, 12, 16, 26, 0, 66, 104, 210, 0, 620, 970, 1748, 0, 5948, 10480, 18976, 0, 60836, 111430, 209460, 0, 704934, 1284836, 2387758, 0, 8331820, 15525814, 28987902, 0, 101242982, 190267598, 358969426, 0, 1275032260
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OFFSET
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1,4
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COMMENT
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Equivalently, number of partitions of the set of the first n
triangular numbers {t(1),..,t(n)} into two classes with equal sums.
Constant term in the expansion of (x+ 1/x)(x^3 +1/x^3)..(x^t(n) +1/x^t(n)).
a(n)=0 for all n=1 (mod 4).
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FORMULA
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Integral representation:
a(n)=(2^n/pi)int_0^pi cos(x)cos(3x)..cos(n(n+1)x/2)dx
Asymptotic formula:
a(n)=2^(n+1)sqrt(10/pi)n^(-5/2)(1+o(1))m as n -->infinity, n not=1 (mod 4).
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EXAMPLE
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For n=6 the 2 solutions are:
+1-3+6-10+15-21=0
-1+3-6+10-15+21=0.
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MAPLE
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N:=70: p:=1: a:=[]: for n from 1 to N do
p:=expand(p*(x^(n*(n+1)/2)+x^(-n*(n+1)/2))):
a:=[op(a), coeff(p, x, 0)]: od:a;
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CROSSREFS
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A063865, A158092, A158118
Sequence in context: A078729 A029906 A094907 this_sequence A051734 A157898 A137430
Adjacent sequences: A158377 A158378 A158379 this_sequence A158381 A158382 A158383
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KEYWORD
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easy,nonn
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AUTHOR
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Pietro Majer (majer(AT)dm.unipi.it), Mar 17 2009
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