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%I #27 Sep 08 2022 08:45:09
%S 1,5,20,70,224,672,1920,5280,14080,36608,93184,232960,573440,1392640,
%T 3342336,7938048,18677760,43581440,100925440,232128512,530579456,
%U 1205862400,2726297600,6134169600,13740539904,30651973632,68115496960
%N a(n) = 2^(n-3)*(n+2)*(n+3)*(n+4)/3.
%C Old definition was "Sequence associated with recurrence a(n)=2*a(n-1)+k(k+2)*a(n-2)". See the first comment in A080928.
%C The fourth column of triangle A080928 (after 0) is 4*a(n).
%H Vincenzo Librandi, <a href="/A080930/b080930.txt">Table of n, a(n) for n = 0..300</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F G.f.: (1-x)*(1-2*x+2*x^2)/(1-2*x)^4 = (1-3*x+4*x^2-2*x^3)/(1-2*x)^4.
%F a(n) = binomial(n+3,3)*2^(n-3), n>0. - _Zerinvary Lajos_, Oct 29 2006
%F a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n>3, a(0)=1, a(1)=5, a(2)=20, a(3)=70. - _Bruno Berselli_, Aug 06 2013
%F E.g.f.: (3 +9*x +6*x^2 +x^3)*exp(2*x)/3. - _G. C. Greubel_, Aug 27 2019
%F From _Amiram Eldar_, Jan 07 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 48*log(2) - 32.
%F Sum_{n>=0} (-1)^n/a(n) = 176 - 432*log(3/2). (End)
%p [seq (binomial(n+3,3)*2^(n-3),n=1..27)]; # _Zerinvary Lajos_, Oct 29 2006
%t CoefficientList[Series[(1-x)(1 -2x +2x^2)/(1-2x)^4, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 06 2013 *)
%t LinearRecurrence[{8, -24, 32, -16}, {1, 5, 20, 70}, 30] (* _Bruno Berselli_, Aug 06 2013 *)
%o (Magma) [Binomial(n+3,3)*2^(n-3): n in [1..30]]; // _Vincenzo Librandi_, Aug 06 2013
%o (PARI) a(n)=2^(n-3)*(n+2)*(n+3)*(n+4)/3 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Sage) [2^(n-2)*binomial(n+4,3) for n in (0..30)] # _G. C. Greubel_, Aug 27 2019
%o (GAP) List([0..30], n-> 2^(n-2)*Binomial(n+4,3)); # _G. C. Greubel_, Aug 27 2019
%Y Cf. A080928.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 26 2003
%E Edited by _Bruno Berselli_, Aug 06 2013
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