A081895 Second binomial transform of binomial(n+3, 3).

1, 6, 30, 136, 579, 2358, 9288, 35640, 133893, 494262, 1797714, 6456024, 22930695, 80660934, 281309436, 973599912, 3346483977, 11431295910, 38828142342, 131206405608, 441271936971, 1477621745046, 4927988620080, 16373939547096

Offset: 0

Author

Paul Barry, Mar 30 2003

Keywords

• nonn
• easy

Comments

Binomial transform of A049612.

2nd binomial transform of binomial(n+3, 3), A000292.

3rd 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 (12,-54,108,-81)

Crossrefs

Cf. A081896.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^3/(1-3*x)^4)); // G. C. Greubel, Oct 18 2018

Mathematica

LinearRecurrence[{12, -54, 108, -81}, {1, 6, 30, 136}, 50] (* G. C. Greubel, Oct 18 2018 *)

PARI

x='x+O('x^30); Vec((1-2*x)^3/(1-3*x)^4) \\ G. C. Greubel, Oct 18 2018

Formula

a(n) = 3^n*(n^3 + 24*n^2 + 137*n + 162)/162.

G.f.: (1 - 2*x)^3/(1 - 3*x)^4.

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