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A232600
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a(n) = Sum_{k=0..n} k^p*q^k, where p=1, q=-2.
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12
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0, -2, 6, -18, 46, -114, 270, -626, 1422, -3186, 7054, -15474, 33678, -72818, 156558, -334962, 713614, -1514610, 3203982, -6757490, 14214030, -29826162, 62448526, -130489458, 272163726, -566697074, 1178133390, -2445745266, 5070447502, -10498808946, 21713445774
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = 2*( (3*n+1)*(-2)^n - 1 )/9.
G.f.: -2*x / ((1 - x)*(1 + 2*x)^2). [corrected by Georg Fischer, May 11 2019]
a(n) = -3*a(n-1) +4*a(n-3). (End)
E.g.f.: (2/9)*(-exp(x) + (1-6*x)*exp(-2*x)).
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EXAMPLE
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a(3) = 0^1*2^0 - 1^1*2^1 + 2^1*2^2 - 3^1*2^3 = -18.
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MAPLE
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MATHEMATICA
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Table[2((3n+1)(-2)^n -1)/9, {n, 0, 30}] (* Bruno Berselli, Nov 28 2013 *)
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PROG
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(PARI) a(n)=-((3*n+1)*(-2)^(n+1)+2)/9;
(Magma) [2*((-2)^n*(3*n+1) -1)/9: n in [0..30]]; // G. C. Greubel, Mar 31 2021
(Sage) [2*((-2)^n*(3*n+1) -1)/9 for n in (0..30)] # G. C. Greubel, Mar 31 2021
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CROSSREFS
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Cf. A045883, A140960 (absolute values), A059841 (p=0, q=-1), A130472 (p=1 ,q=-1), A089594 (p=2, q=-1), A232599 (p=3, q=-1), A126646 (p=0, q=2), A036799 (p=1, q=2), A036800 (p=q=2), A036827 (p=3, q=2), A077925 (p=0, q=-2), A232601 (p=2, q=-2), A232602 (p=3, q=-2), A232603 (p=2, q=-1/2), A232604 (p=3, q=-1/2).
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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