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Third binomial transform of C(n+2,2).
3

%I #14 Sep 08 2022 08:45:09

%S 1,6,33,172,864,4224,20224,95232,442368,2031616,9240576,41680896,

%T 186646528,830472192,3674210304,16173236224,70866960384,309237645312,

%U 1344324763648,5823975653376,25151328485376,108301895335936

%N Third binomial transform of C(n+2,2).

%C Binomial transform of A081892.

%C 3rd binomial transform of C(n+2,2), A000217.

%C 4th binomial transform of (1,2,1,0,0,0,.....)

%H G. C. Greubel, <a href="/A081893/b081893.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-48,64).

%F a(n) = 4^n*(n^2 + 15*n + 32)/32.

%F G.f.: (1 - 3*x)^2/(1 - 4*x)^3.

%F E.g.f.: (2 + 4*x + x^2)*exp(4*x)/2. - _G. C. Greubel_, Oct 18 2018

%t LinearRecurrence[{12, -48, 64}, {1, 6, 33}, 50] (* _G. C. Greubel_, Oct 18 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-3*x)^2/(1-4*x)^3) \\ _G. C. Greubel_, Oct 18 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x)^2/(1-4*x)^3)); // _G. C. Greubel_, Oct 18 2018

%Y Cf. A081894.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 30 2003