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A141478 a(n) = binomial(n+2,3)*4^3.

%I A141478 #35 Aug 19 2025 16:38:40
%S A141478 64,256,640,1280,2240,3584,5376,7680,10560,14080,18304,23296,29120,
%T A141478 35840,43520,52224,62016,72960,85120,98560,113344,129536,147200,
%U A141478 166400,187200,209664,233856,259840,287680,317440,349184,382976,418880,456960,497280,539904,584896
%N A141478 a(n) = binomial(n+2,3)*4^3.
%H A141478 Vincenzo Librandi, <a href="/A141478/b141478.txt">Table of n, a(n) for n = 1..1000</a>
%H A141478 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A141478 G.f.: 64*x/(1-x)^4.
%F A141478 a(n) = 32*n*(n+1)*(n+2)/3 = 64*A000292(n).
%F A141478 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 29 2012
%F A141478 From _Amiram Eldar_, Aug 29 2022: (Start)
%F A141478 Sum_{n>=1} 1/a(n) = 3/128.
%F A141478 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)/16 - 15/128. (End)
%F A141478 From _Elmo R. Oliveira_, Aug 19 2025: (Start)
%F A141478 E.g.f.: 32*x*(6 + 6*x + x^2)*exp(x)/3.
%F A141478 a(n) = 16*A210440(n). (End)
%p A141478 seq(binomial(n+2,3)*4^3, n=1..36);
%t A141478 CoefficientList[Series[64/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 29 2012 *)
%o A141478 (Magma) [Binomial(n+2,3)*4^3: n in [1..34]];  // _Bruno Berselli_, Apr 07 2011
%o A141478 (Magma) I:=[64, 256, 640, 1280]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 29 2012
%Y A141478 Cf. A000292, A035008, A038231 (3rd subdiagonal), A210440.
%K A141478 nonn,easy
%O A141478 1,1
%A A141478 _Zerinvary Lajos_, Aug 09 2008
%E A141478 Offset adapted to the g.f. by _Bruno Berselli_, Apr 07 2011