OFFSET
0,2
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 - 3*n^2 + 2*n + 6*k^3)/(6*k^3), with g.f. (1 - 3*k*x + 3*k^2*x^2 - (k^3-1)*x^3)/(1-k*x)^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).
FORMULA
a(n) = 5^n*(n^3 - 3*n^2 + 2*n + 750)/750.
G.f.: (1 - 15*x + 75*x^2 - 124*x^3)/(1-5*x)^4.
From Elmo R. Oliveira, Nov 12 2025: (Start)
E.g.f.: (1 + x^3/6)*exp(5*x).
a(n) = 20*a(n-1) - 150*a(n-2) + 500*a(n-3) - 625*a(n-4). (End)
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 *)
PROG
(Magma) [5^n*(n^3-3*n^2+2*n+750)/750: n in [0..40]]; // Vincenzo Librandi, Apr 27 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 31 2003
STATUS
approved