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A100381
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a(n) = 2^n*binomial(n,2).
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5
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0, 0, 4, 24, 96, 320, 960, 2688, 7168, 18432, 46080, 112640, 270336, 638976, 1490944, 3440640, 7864320, 17825792, 40108032, 89653248, 199229440, 440401920, 968884224, 2122317824, 4630511616, 10066329600, 21810380800, 47110422528
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OFFSET
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0,3
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COMMENTS
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a(n) is the number of ways to partition the set [n]={1,2,...,n} into two sets S,T and select 2 elements in total (from either S or T or both).
Example. For n=4, sample partitions are given (where S(i),T(j) means i elements are selected from S, j elements are selected from T):
S={ }, T={1,2,3,4}: partition [4] in 1 way, S(0),T(2) (6 ways);
S={1}, T={2,3,4}: partition [4] in 4 such ways, S(1),T(1) or S(0),T(2) (24 ways);
S={1,2}, T={3,4}: partition [4], in such 6 ways, S(1),T(1) or S(0),T(2) or S(2),T(0) (36 ways);
S={1,2,3}, T={4}: partition [4] in 4 such ways, S(1),T(1) or S(2),T(0) (24 ways);
S={1,2,3,4}, T={ }: partition [4] in 1 way, S(2),T(0) (6 ways). (End)
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REFERENCES
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Jolley, Summation of Series, Dover (1961), eq (214) page 40.
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LINKS
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FORMULA
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Sum_{n>=2} 1/a(n) = 1 - log(2) = 0.3068528.... - Graeme McRae, Jul 28 2006
E.g.f.: 2*x^2*exp(2x).
Sum_{j=1..k} (j+2)/a(j+1) = 1 - 1/((k+1)*2^k). [Jolley]
Sum_{n>=2} (-1)^n/a(n) = 3*log(3/2) - 1. - Amiram Eldar, Jul 20 2020
Sum_{k = 2..n+2} 1/a(k) = 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n + 1)/(3*n + 4)))).
Sum_{k = 2..n+2} (-1)^k/a(k) = 1/(4 + 4/(5 + 12/(6 + ... + 2*n*(n + 1)/(n + 4)))).
Letting n -> oo in the above gives the continued fraction representations
1 - log(2) = Sum_{k >= 2} 1/a(k) = 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n + 1)/((3*n + 4) - ... )))) (an equivalent continued fraction for 1 - log(2) was conjectured by the Ramanujan machine) and
3*log(3/2) - 1 = Sum_{k >= 2} (-1)^k/a(k) = 1/(4 + 4/(5 + 12/(6 + ... + 2*n*(n + 1)/((n + 4) + ... )))). (End)
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MAPLE
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seq(2^n*binomial(n, 2), n=0..20);
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MATHEMATICA
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Range[0, 20]! CoefficientList[Series[2x^2 Exp[2x], {x, 0, 20}], x]
Table[2^n Binomial[n, 2], {n, 0, 30}] (* or *) LinearRecurrence[{6, -12, 8}, {0, 0, 4}, 30] (* Harvey P. Dale, Aug 15 2020 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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