%I #17 Sep 08 2022 08:45:09
%S 1,7,43,245,1328,6944,35328,175872,860160,4145152,19726336,92864512,
%T 433061888,2002780160,9193914368,41926262784,190052302848,
%U 856845975552,3843995729920,17166984282112,76347338653696,338237264494592
%N A sequence related to binomial(n+3, 3).
%C Binomial transform of A081895.
%C 3rd binomial transform of binomial(n+3, 3), A000292.
%C 4th binomial transform of (1,3,3,1,0,0,0,0,...).
%H G. C. Greubel, <a href="/A081896/b081896.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (16,-96,256,-256)
%F a(n) = 4^n*(n^3 + 33*n^2 + 254*n + 384)/384.
%F G.f.: (1 - 3*x)^3/(1 - 4*x)^4.
%F E.g.f.: (6 + 18*x + 9*x^2 + x^3)*exp(4*x)/6. - _G. C. Greubel_, Oct 18 2018
%t LinearRecurrence[{16, -96, 256, -256}, {1, 7, 43, 245}, 50] (* _G. C. Greubel_, Oct 18 2018 *)
%t CoefficientList[Series[(1-3x)^3/(1-4x)^4,{x,0,30}],x] (* _Harvey P. Dale_, Nov 30 2021 *)
%o (PARI) x='x+O('x^30); Vec((1-3*x)^3/(1-4*x)^4) \\ _G. C. Greubel_, Oct 18 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^3/(1-4*x)^4)); // _G. C. Greubel_, Oct 18 2018
%Y Cf. A081897.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 30 2003