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

0,4

Comments

Binomial transform is A105367. Consecutive pair sums of 105369.

Links

Table of n, a(n) for n=0..38.

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

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) From Harvey P. Dale, May 23 2012

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 *)

Crossrefs

Sequence in context: A352189 A101725 A274453 * A173694 A190459 A135042

Adjacent sequences: A105367 A105368 A105369 * A105371 A105372 A105373

Keyword

easy,sign

Author

Paul Barry, Apr 01 2005

Status

approved