A080929 Sequence associated with a(n) = 2*a(n-1) + k*(k+2)*a(n-2).
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
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
Crossrefs
Programs
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GAP
Concatenation([1], List([1..30], n-> 2^(n-1)*Binomial(n+2,2))); # G. C. Greubel, Jul 23 2019
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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
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Maple
[seq (ceil(binomial(n+2,2)*2^(n-1)),n=0..30)]; # Zerinvary Lajos, Nov 01 2006
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Mathematica
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 *) -
PARI
vector(30, n, n--; if(n==0,1, 2^(n-1)*binomial(n+2,2) )) \\ G. C. Greubel, Jul 23 2019
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Sage
[1]+[2^(n-1)*binomial(n+2,2) for n in (1..30)] # G. C. Greubel, Jul 23 2019
Formula
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
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 7 - 8*log(2).
Sum_{n>=0} (-1)^n/a(n) = 24*log(3/2) - 9. (End)
Comments