A081897 Fourth binomial transform of binomial(n+3, 3).

1, 8, 58, 396, 2595, 16500, 102500, 625000, 3753125, 22250000, 130468750, 757812500, 4365234375, 24960937500, 141796875000, 800781250000, 4498291015625, 25146484375000, 139953613281250, 775756835937500, 4283905029296875, 23574829101562500, 129318237304687500, 707244873046875000

Offset: 0

Author

Paul Barry, Mar 30 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A081895.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (20,-150,500,-625)

Crossrefs

Cf. A000292, A081895.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)^3/(1-5*x)^4)); // G. C. Greubel, Oct 18 2018

Mathematica

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

PARI

x='x+O('x^30); Vec((1-4*x)^3/(1-5*x)^4) \\ G. C. Greubel, Oct 18 2018

Formula

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

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

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