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%I #24 Sep 08 2022 08:45:09
%S 1,5,30,140,560,2016,6720,21120,63360,183040,512512,1397760,3727360,
%T 9748480,25067520,63504384,158760960,392232960,958791680,2321285120,
%U 5571084288,13264486400,31352422400,73610035200,171756748800
%N Sequence associated with recurrence a(n) = 2*a(n-1) + k*(k+2)*a(n-2).
%C Fifth column of triangle A080928.
%H Vincenzo Librandi, <a href="/A080951/b080951.txt">Table of n, a(n) for n = 0..400</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F G.f.: (1-x)*(1 - 4*x + 16*x^2 - 24*x^3 + 16*x^4)/(1-2*x)^5.
%F a(n) = ceiling(binomial(n+4,4)*2^(n-1)). - _Zerinvary Lajos_, Nov 01 2006
%F From _Amiram Eldar_, Jan 07 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 37/3 - 16*log(2).
%F Sum_{n>=0} (-1)^n/a(n) = 432*log(3/2) - 523/3. (End)
%p [seq( ceil(binomial(n+4,4)*2^(n-1)),n=0..30)]; # _Zerinvary Lajos_, Nov 01 2006
%t Join[{1}, LinearRecurrence[{10,-40,80,-80,32}, {5,30,140,560,2016}, 30]] (* _G. C. Greubel_, Jul 23 2019 *)
%o (Magma) [(Ceiling(Binomial(n+4, 4)*2^(n-1))) : n in [0..30]]; // _Vincenzo Librandi_, Sep 22 2011
%o (PARI) my(x='x+O('x^30)); Vec((1-x)*(1-4*x+16*x^2-24*x^3 +16*x^4)/(1 -2*x)^5) \\ _G. C. Greubel_, Jul 23 2019
%o (Sage) ((1-x)*(1-4*x+16*x^2-24*x^3+16*x^4)/(1-2*x)^5).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 23 2019
%o (GAP) a:=[5,30,140,560,2016];; for n in [6..30] do a[n]:=10*a[n-1] -40*a[n-2]+80*a[n-3]-80*a[n-4]+32*a[n-5]; od; Concatenation([1], a); # _G. C. Greubel_, Jul 23 2019
%Y Cf. A082140, A082141, A082138, A082139, A080928, A080929, A057711.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 26 2003
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