A189376 Expansion of 1/((1-x)^5*(x^3+x^2+x+1)^2).

1, 3, 6, 10, 17, 27, 40, 56, 78, 106, 140, 180, 230, 290, 360, 440, 535, 645, 770, 910, 1071, 1253, 1456, 1680, 1932, 2212, 2520, 2856, 3228, 3636, 4080, 4560, 5085, 5655, 6270, 6930, 7645, 8415, 9240, 10120, 11066, 12078

Offset: 0

Author

Johannes W. Meijer, Apr 29 2011

Keywords

• easy
• nonn

Comments

The Gi2 triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of the Gi2 and other triangle sums see A180662.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1,2,-6,6,-2,-1,3,-3,1).

Crossrefs

Cf. A139600, A189374, A189375.

Programs

Maple

a:= n-> coeff (series (1/((1-x)^5*(x^3+x^2+x+1)^2), x, n+1), x, n):
seq (a(n), n=0..50);

Mathematica

CoefficientList[Series[1/((1-x)^5(x^3+x^2+x+1)^2),{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1,2,-6,6,-2,-1,3,-3,1},{1,3,6,10,17,27,40,56,78,106,140},50] (* Harvey P. Dale, Apr 12 2015 *)

Formula

a(n) = sum(A144678(n-k), k=0..n).

Gi2(n) = A189376(n-1) - A189376(n-2) - A189376(n-5) + 2*A189376(n-6) with A189376(n)=0 for n <= -1.

a(0)=1, a(1)=3, a(2)=6, a(3)=10, a(4)=17, a(5)=27, a(6)=40, a(7)=56, a(8)=78, a(9)=106, a(10)=140, a(n)=3*a(n-1)-3*a(n-2)+a(n-3)+ 2*a(n-4)- 6*a(n-5)+6*a(n-6)-2*a(n-7)-a(n-8)+3*a(n-9)-3*a(n-10)+a(n-11). - Harvey P. Dale, Apr 12 2015