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A052936 Expansion of (1-x)(1-2x)/(1-5x+5x^2). 2
1, 2, 7, 25, 90, 325, 1175, 4250, 15375, 55625, 201250, 728125, 2634375, 9531250, 34484375, 124765625, 451406250, 1633203125, 5908984375, 21378906250, 77349609375, 279853515625, 1012519531250, 3663330078125, 13254052734375 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n>=0, a(n) is the number of generalized compositions of n+1 when there are 2^(i-1)+2 different types of i, (i=1,2,...). [From Milan Janjic, Sep 24 2010]

LINKS

Table of n, a(n) for n=0..24.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 925

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

FORMULA

G.f.: (-1+x)*(-1+2*x)/(1-5*x+5*x^2)

a(0)=1, a(1)=2, a(2)=7, a(n)=5*a(n-1)-5*a(n-2) [From Harvey P. Dale, Apr 20 2012]

Sum(-1/5*(-1+_alpha)*_alpha^(-1-n), _alpha=RootOf(1-5*_Z+5*_Z^2))

The sequence beginning 2, 7, 25 ... has g.f. (2-3x)/(1-5x+5x^2), a(n)=(1-2/sqrt(5))(5/2-sqrt(5)/2)^n+(5/2+sqrt(5)/2)^n(1+2/sqrt(5)). It is the binomial transform of Fib(2n+3) and the second binomial transform of Fib(n+3). Also, its n-th term is the n-th term of the 3rd binomial transform of Fib(3n+3) divided by 2^n. - Paul Barry, Mar 23 2004

Binomial transform of convolution of Fib(2n+1) and (-1)^n. Binomial transform of Fib(n+1)^2. - Paul Barry, Sep 27 2004

a(n)=sum{k=0..n, C(n-1, k)(Fib(2n-2k)+Fib(2n-2k-1))} - Paul Barry, Jun 07 2005

MAPLE

spec := [S, {S=Sequence(Prod(Union(Sequence(Z), Sequence(Union(Z, Z))), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

MATHEMATICA

Join[{1}, LinearRecurrence[{5, -5}, {2, 7}, 30]] (* Harvey P. Dale, Apr 20 2012 *)

CROSSREFS

Sequence in context: A070859 A048576 A018907 * A108152 A024482 A097613

Adjacent sequences:  A052933 A052934 A052935 * A052937 A052938 A052939

KEYWORD

easy,nonn

AUTHOR

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

EXTENSIONS

More terms from James A. Sellers, Jun 06 2000

STATUS

approved

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Last modified January 17 14:12 EST 2019. Contains 319225 sequences. (Running on oeis4.)