A189374 Expansion of 1/((1-x)^5*(x^2+x+1)^3).

1, 2, 3, 7, 11, 15, 25, 35, 45, 65, 85, 105, 140, 175, 210, 266, 322, 378, 462, 546, 630, 750, 870, 990, 1155, 1320, 1485, 1705, 1925, 2145, 2431, 2717, 3003, 3367, 3731, 4095, 4550, 5005, 5460, 6020, 6580, 7140, 7820

Offset: 0

Author

Johannes W. Meijer, Apr 29 2011

Keywords

• easy
• nonn

Comments

The Ca1(n) and Ze3(n) triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of these triangle sums see A180662.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,3,-6,3,-3,6,-3,1,-2,1)

Crossrefs

Cf. A139600, A189375, A189376.

Programs

Maple

a:= proc(n) option remember; `if` (n<4, [1, 2, 3, 7][n+1], (2*a(n-1) +2*a(n-2) +(8+n) *a(n-3))/n) end: seq (a(n), n=0..50);

Formula

a(n) = (2*a(n-1) + 2*a(n-2) + (8+n)*a(n-3))/n with a(0)=1, a(1)=2, a(2)=3 and a(3)=7.

a(n) = sum(A011779(n-k)*A049347(k), k=0..n).

Ca1(n) = A189374(n-3) - A189374(n-4) - A189374(n-6) + 2*A189374(n-7).

Ze3(n) = 2*A189374(n-3) - A189374(n-4) - 2*A189374(n-6) + 5*A189374(n-7) with A189374(n)=0 for n <= -1.

a(n) = (floor(n/3)+1)*(floor(n/3)+2)*(floor(n/3)+3)*(3*floor(n/3)+4*(4-(3*floor((n+3)/3)-n)))/24. - Luce ETIENNE, Jun 29 2015