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

%I #12 Sep 08 2022 08:45:09

%S 1,7,46,290,1775,10625,62500,362500,2078125,11796875,66406250,

%T 371093750,2060546875,11376953125,62500000000,341796875000,

%U 1861572265625,10101318359375,54626464843750,294494628906250,1583099365234375

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

%C Binomial transform of A081893.

%C 4th binomial transform of C(n+2,2), A000217.

%C 5th binomial transform of (1,2,1,0,0,0,.....)

%H G. C. Greubel, <a href="/A081894/b081894.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 (15,-75,125).

%F a(n) = 5^n*(n^2 + 19*n + 50)/50.

%F G.f.: (1 - 4*x)^2/(1 - 5*x)^3.

%F E.g.f.: (2 + 4*x + x^2)*exp(5*x)/2. - _G. C. Greubel_, Oct 18 2018

%t LinearRecurrence[{15, -75, 125}, {1, 7, 46}, 50] (* _G. C. Greubel_, Oct 18 2018 *)

%o (PARI) x='x+O('x^30); Vec((1-4*x)^2/(1-5*x)^3) \\ _G. C. Greubel_, Oct 18 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)^2/(1-5*x)^3)); // _G. C. Greubel_, Oct 18 2018

%Y Cf. A081907.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 30 2003