|
%I #14 Sep 08 2022 08:45:09
%S 1,6,30,136,579,2358,9288,35640,133893,494262,1797714,6456024,
%T 22930695,80660934,281309436,973599912,3346483977,11431295910,
%U 38828142342,131206405608,441271936971,1477621745046,4927988620080,16373939547096
%N Second binomial transform of binomial(n+3, 3).
%C Binomial transform of A049612.
%C 2nd binomial transform of binomial(n+3, 3), A000292.
%C 3rd binomial transform of (1,3,3,1,0,0,0,0,...).
%H G. C. Greubel, <a href="/A081895/b081895.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 (12,-54,108,-81)
%F a(n) = 3^n*(n^3 + 24*n^2 + 137*n + 162)/162.
%F G.f.: (1 - 2*x)^3/(1 - 3*x)^4.
%F E.g.f.: (6 + 18*x + 9*x^2 + x^3)*exp(3*x)/6. - _G. C. Greubel_, Oct 18 2018
%t LinearRecurrence[{12, -54, 108, -81}, {1, 6, 30, 136}, 50] (* _G. C. Greubel_, Oct 18 2018 *)
%o (PARI) x='x+O('x^30); Vec((1-2*x)^3/(1-3*x)^4) \\ _G. C. Greubel_, Oct 18 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^3/(1-3*x)^4)); // _G. C. Greubel_, Oct 18 2018
%Y Cf. A081896.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 30 2003
|
