A081916 a(n) = 5^n*(n^3 - 3n^2 + 2n + 750)/750.

1, 5, 25, 126, 645, 3375, 18125, 100000, 565625, 3265625, 19140625, 113281250, 673828125, 4013671875, 23876953125, 141601562500, 836181640625, 4913330078125, 28717041015625, 166931152343750, 965118408203125

Offset: 0

Author

Paul Barry, Mar 31 2003

Keywords

• easy
• nonn

Comments

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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..150
• Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625).

Programs

Magma

[5^n*(n^3-3*n^2+2*n+750)/750: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011

Mathematica

a[n_]:= 5^n*(n^3 - 3n^2 + 2n + 750)/750 ; Array[a, 40, 0] (* or *)
CoefficientList[Series[(1 - 15x + 75x^2 - 124x^3)/(1-5x)^4 ,{x, 0, 40}], x] (* Stefano Spezia, Sep 02 2018 *)
LinearRecurrence[{20,-150,500,-625},{1,5,25,126},30] (* Harvey P. Dale, Jun 29 2021 *)

Formula

a(n) = 5^n*(n^3 - 3n^2 + 2n + 750)/750.

G.f.: (1 - 15x + 75x^2 - 124x^3)/(1-5x)^4.