A190050 Expansion of ((1-x)*(3*x^2-3*x+1))/(1-2*x)^3.

1, 2, 6, 17, 46, 120, 304, 752, 1824, 4352, 10240, 23808, 54784, 124928, 282624, 634880, 1417216, 3145728, 6946816, 15269888, 33423360, 72876032, 158334976, 342884352, 740294656, 1593835520, 3422552064

Offset: 0

Author

Johannes W. Meijer, May 06 2011

Keywords

• nonn
• easy

Comments

The second left hand column of triangle A175136.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

Crossrefs

Cf. A175136, A011782, A190951.

Related to A001788.

Programs

Magma

[1] cat [(n^2 + 5*n + 10)*2^(n-4): n in [1..30]]; // G. C. Greubel, Jan 10 2018

Maple

A190050:= proc(n) option remember; if n=0 then A190050(n):=1: else A190050(n):=(n^2+5*n+10)*2^(n-4) fi: end: seq (A190050(n), n=0..26);

Mathematica

Join[{1}, LinearRecurrence[{6,-12,8}, {2,6,17}, 30]] (* or *) CoefficientList[Series[((1-x)*(3*x^2-3*x+1))/(1-2*x)^3, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

PARI

x='x+O('x^30); Vec(((1-x)*(3*x^2-3*x+1))/(1-2*x)^3) \\ G. C. Greubel, Jan 10 2018

PARI

for(n=0,30, print1(if(n==0,1,(n^2 + 5*n + 10)*2^(n-4)), ", ")) \\ G. C. Greubel, Jan 10 2018

Formula

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

a(n) = (n^2 + 5*n + 10)*2^(n-4) for n >=1 with a(0)=1.

a(n) = A001788(n+1) -4*A001788(n) +6*A001788(n-1) -3*A001788(n-2) for n >=1 with a(0)=1.