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%I #16 Sep 08 2022 08:45:09
%S 1,5,25,126,645,3375,18125,100000,565625,3265625,19140625,113281250,
%T 673828125,4013671875,23876953125,141601562500,836181640625,
%U 4913330078125,28717041015625,166931152343750,965118408203125
%N a(n) = 5^n*(n^3 - 3n^2 + 2n + 750)/750.
%C Binomial transform of A081915. 5th binomial transform of (1,0,0,1,0,0,0,0,...). Case k=5 where a(n,k) = k^n*(n^3 - 3n^2 + 2n + 6k^3)/(6k^3), with g.f. (1 - 3kx + 3k^2x^2 - (k^3-1)x^3)/(1-kx)^4.
%H Vincenzo Librandi, <a href="/A081916/b081916.txt">Table of n, a(n) for n = 0..150</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 - 3n^2 + 2n + 750)/750.
%F G.f.: (1 - 15x + 75x^2 - 124x^3)/(1-5x)^4.
%t a[n_]:= 5^n*(n^3 - 3n^2 + 2n + 750)/750 ; Array[a, 40, 0] (* or *)
%t CoefficientList[Series[(1 - 15x + 75x^2 - 124x^3)/(1-5x)^4 ,{x, 0, 40}], x] (* _Stefano Spezia_, Sep 02 2018 *)
%t LinearRecurrence[{20,-150,500,-625},{1,5,25,126},30] (* _Harvey P. Dale_, Jun 29 2021 *)
%o (Magma) [5^n*(n^3-3*n^2+2*n+750)/750: n in [0..40]]; // _Vincenzo Librandi_, Apr 27 2011
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 31 2003