A226405 Expansion of x/((1-x-x^3)*(1-x)^3).

0, 1, 4, 10, 21, 40, 71, 120, 196, 312, 487, 749, 1139, 1717, 2571, 3830, 5683, 8407, 12408, 18281, 26898, 39537, 58071, 85245, 125082, 183478, 269074, 394534, 578418, 847927, 1242926, 1821840, 2670295, 3913782, 5736217, 8407142, 12321590, 18058510, 26466393

Offset: 0

Author

Alois P. Heinz, Jun 06 2013

Keywords

• nonn
• easy

Comments

From Bruno Berselli, Jun 07 2013: (Start)

A050228(n) = a(n) -a(n-1), n>0.

A077868(n-1)= a(n) -2*a(n-1) +a(n-2), n>1.

A000217(n) = a(n) -a(n-1) -a(n-3), n>2.

A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2) -a(n-3), n>2.

n = a(n) -2*a(n-1) +a(n-2) -a(n-3) +a(n-4), n>3.

1 = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4) -a(n-5), n>4.

0 = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.

(End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-4,3,-1).

Crossrefs

4th column of A144903.

Cf. A000217, A078012, A099567, A135851.

Cf. A000930, A050228, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

Programs

Magma

A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
[A226405(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

Maple

a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

Mathematica

LinearRecurrence[{4,-6,5,-4,3,-1}, {0,1,4,10,21,40}, 40]  (* Bruno Berselli, Jun 07 2013 *)
CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

PARI

my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ G. C. Greubel, Apr 28 2017

SageMath

def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
[A226405(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

Formula

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

From G. C. Greubel, Jul 27 2022: (Start)

a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).

a(n) = A099567(n+2, 3). (End)