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A sequence related to binomial(n+3, 3).
2

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

%S 1,8,58,396,2595,16500,102500,625000,3753125,22250000,130468750,

%T 757812500,4365234375,24960937500,141796875000,800781250000,

%U 4498291015625,25146484375000,139953613281250,775756835937500

%N A sequence related to binomial(n+3, 3).

%C Binomial transform of A081895.

%C 4th binomial transform of binomial(n+3, 3), A000292.

%C 5th binomial transform of (1,3,3,1,0,0,0,0,...).

%H G. C. Greubel, <a href="/A081897/b081897.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 (20,-150,500,-625)

%F a(n) = 5^n*(n^3 + 42*n^2 + 407*n + 750)/750.

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

%F E.g.f.: (6 + 18*x + 9*x^2 + x^3)*exp(5*x)/6. - _G. C. Greubel_, Oct 18 2018

%t LinearRecurrence[{20, -150, 500, -625}, {1, 8, 58, 396}, 50] (* _G. C. Greubel_, Oct 18 2018 *)

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

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

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 30 2003