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A080929
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Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).
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12
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1, 3, 12, 40, 120, 336, 896, 2304, 5760, 14080, 33792, 79872, 186368, 430080, 983040, 2228224, 5013504, 11206656, 24903680, 55050240, 121110528, 265289728, 578813952, 1258291200, 2726297600, 5888802816, 12683575296, 27246198784
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OFFSET
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0,2
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COMMENTS
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The third column of number triangle A080928.
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LINKS
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FORMULA
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G.f.: (1-x)*(1-2*x+4*x^2)/(1-2*x)^3.
For n>0, a(n) = (n+1)*(n+2)*2^(n-2). - Ralf Stephan, Jan 16 2004
a(n) = Sum_{k=0..n} Sum_{i=0..n} (k+1)*binomial(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
Sum_{n>=0} 1/a(n) = 7 - 8*log(2).
Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)
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MAPLE
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[seq (ceil(binomial(n+2, 2)*2^(n-1)), n=0..30)]; # Zerinvary Lajos, Nov 01 2006
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MATHEMATICA
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CoefficientList[Series[(1-x)(1-2x+4x^2)/(1-2x)^3, {x, 0, 30}], x] (* Michael De Vlieger, Sep 21 2017 *)
Join[{1}, LinearRecurrence[{6, -12, 8}, {3, 12, 40}, 30]] (* G. C. Greubel, Jul 23 2019 *)
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PROG
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(Magma) [n eq 0 select 1 else (n+1)*(n+2)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Sep 22 2011
(PARI) vector(30, n, n--; if(n==0, 1, 2^(n-1)*binomial(n+2, 2) )) \\ G. C. Greubel, Jul 23 2019
(Sage) [1]+[2^(n-1)*binomial(n+2, 2) for n in (1..30)] # G. C. Greubel, Jul 23 2019
(GAP) Concatenation([1], List([1..30], n-> 2^(n-1)*Binomial(n+2, 2))); # G. C. Greubel, Jul 23 2019
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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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