A105370 Expansion of ((1+x)^4-(1+x)x^3)/((1+x)^5-x^5).

1, -1, 1, -2, 5, -10, 15, -15, 0, 50, -175, 450, -1000, 2000, -3625, 5875, -8125, 8125, 0, -29375, 106250, -278125, 621875, -1243750, 2250000, -3640625, 5031250, -5031250, 0, 18203125, -65859375, 172421875, -385546875, 771093750, -1394921875, 2257031250, -3119140625, 3119140625, 0

Offset: 0

Author

Paul Barry, Apr 01 2005

Keywords

• easy
• sign

Comments

Binomial transform is A105367. Consecutive pair sums of 105369.

Links

• Index entries for linear recurrences with constant coefficients, signature (-5,-10,-10,-5).

Programs

Mathematica

CoefficientList[Series[((1+x)^4-(1+x)x^3)/((1+x)^5-x^5),{x,0,40}],x] (* or *) LinearRecurrence[{-5,-10,-10,-5},{1,-1,1,-2},41]  (* Harvey P. Dale, May 23 2012 *)

Formula

G.f.: (1+x)(1+3x+3x^2)/(1+5x+10x^2+10x^3+5x^4).

a(n) = (5/2-sqrt(5)/2)^(n/2)((1/2+sqrt(5)/10)cos(7*Pi*n/10)+ sqrt(1/10-sqrt(5)/50)sin(7*Pi*n/10))- (5/2+sqrt(5)/2)^(n/2)((sqrt(5)/10-1/2)cos(9*Pi*n/10)+sqrt(1/10+sqrt(5)/50)sin(9*Pi*n/10)).

a(0)=1, a(1)=-1, a(2)=1, a(3)=-2, a(n)=-5*a(n-1)-10*a(n-2)- 10*a(n-3)- 5*a(n-4). - Harvey P. Dale, May 23 2012