A119336 Expansion of (1-x)^4/((1-x)^6 - x^6).

1, 2, 3, 4, 5, 6, 8, 16, 45, 130, 341, 804, 1730, 3460, 6555, 12016, 21845, 40410, 77540, 155080, 320001, 669526, 1398101, 2884776, 5858126, 11716252, 23166783, 45536404, 89478485, 176565486, 350739488, 701478976, 1410132405, 2841788170

Offset: 0

Author

Paul Barry, May 14 2006

Keywords

• easy
• nonn

Comments

Row sums of A119335. Binomial transform of (1+x)/(1-x)^6.

Equals binomial transform of [1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, ...]. - Gary W. Adamson, Mar 14 2009

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Crossrefs

Cf. A119335, A306847.

Programs

Mathematica

CoefficientList[Series[(1-x)^4/((1-x)^6-x^6),{x,0,40}],x] (* or *) LinearRecurrence[{6,-15,20,-15,6},{1,2,3,4,5},40] (* Harvey P. Dale, Dec 25 2015 *)

PARI

{a(n) = sum(k=0, n\6, binomial(n+1, 6*k+1))} \\ Seiichi Manyama, Mar 22 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5), with a(0)=1, a(1)=2, a(2)=3, a(3)=4, a(4)=5. - Harvey P. Dale, Dec 25 2015

a(n) = Sum_{k=0..floor(n/6)} binomial(n+1,6*k+1). - Seiichi Manyama, Mar 22 2019