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A081899 Third binomial transform of binomial(n+4, 4).

%I A081899 #16 Mar 22 2025 12:27:33
%S A081899 1,8,54,332,1921,10644,57072,298176,1525248,7665664,37953536,
%T A081899 185499648,896466944,4289462272,20343422976,95718211584,447146360832,
%U A081899 2075274510336,9574555844608,43933220470784,200579267690496,911512319295488,4124474274217984,18588068701274112
%N A081899 Third binomial transform of binomial(n+4, 4).
%C A081899 Binomial transform of A081898.
%C A081899 4th binomial transform of (1,4,6,4,1,0,0,0,...).
%H A081899 G. C. Greubel, <a href="/A081899/b081899.txt">Table of n, a(n) for n = 0..1000</a>
%H A081899 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (20,-160,640,-1280,1024).
%F A081899 a(n) = 4^n*(n^4 + 58*n^3 + 971*n^2 + 5114*n + 6144)/6144.
%F A081899 G.f.: (1 - 3*x)^4/(1 - 4*x)^5.
%F A081899 E.g.f.: (24 + 96*x + 72*x^2 + 16*x^3 + x^4)*exp(4*x)/24. - _G. C. Greubel_, Oct 18 2018
%t A081899 LinearRecurrence[{20,-160,640,-1280,1024}, {1,8,54,332,1921}, 50] (* _G. C. Greubel_, Oct 18 2018 *)
%o A081899 (PARI) x='x+O('x^30); Vec((1-3*x)^4/(1-4*x)^5) \\ _G. C. Greubel_, Oct 18 2018
%o A081899 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^4/(1-4*x)^5)); // _G. C. Greubel_, Oct 18 2018
%Y A081899 Cf. A000332, A081898, A081900.
%K A081899 nonn,easy
%O A081899 0,2
%A A081899 _Paul Barry_, Mar 30 2003