A052960 Expansion of ( 1-x-x^2 ) / ( 1-2*x-2*x^2+x^3+x^4 ).

1, 1, 3, 7, 18, 46, 118, 303, 778, 1998, 5131, 13177, 33840, 86905, 223182, 573157, 1471933, 3780093, 9707713, 24930522, 64024444, 164422126, 422254905, 1084399096, 2784861432, 7151844025, 18366756913, 47167941348, 121132691065

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Diagonal sums of the Riordan matrix ((1-x-x^2)/(1-2*x-x^2), (x-x^2-x^3) / (1-2*x-x^2)) (A190215). - Emanuele Munarini, May 10 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1031
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1,-1).

Programs

GAP

a:=[1,1,3,7];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Oct 23 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x-x^2)/(1-2*x-2*x^2+x^3+x^4) )); // G. C. Greubel, Oct 23 2019

Maple

spec:= [S,{S=Sequence(Prod(Union(Sequence(Union(Prod(Z,Z),Z)),Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 23 2019

Mathematica

Table[Sum[Sum[Binomial[i+2k,2k]Binomial[i+k,n-i-2k],{k,0,n/2}],{i,0,n}],{n,0,12}] (* Emanuele Munarini, May 10 2011 *)
LinearRecurrence[{2,2,-1,-1}, {1,1,3,7}, 30] (* G. C. Greubel, Oct 23 2019 *)
CoefficientList[Series[(1-x-x^2)/(1-2x-2x^2+x^3+x^4),{x,0,50}],x] (* Harvey P. Dale, Jan 21 2021 *)

Maxima

makelist(sum(sum(binomial(i+2*k, 2*k)*binomial(i+k, n-i-2*k), k,0,n/2),i,0,n),n,0,24); /* Emanuele Munarini, May 10 2011 */

PARI

my(x='x+O('x^30)); Vec((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4)) \\ G. C. Greubel, Oct 23 2019

Sage

def A052960_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x-x^2)/(1-2*x-2*x^2+x^3+x^4)).list()
A052960_list(30) # G. C. Greubel, Oct 23 2019

Formula

G.f.: (1-x-x^2)/(1-2*x-2*x^2+x^3+x^4).

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).

a(n) = Sum_{alpha=RootOf(1-2*z-2*z^2+z^3+z^4)} (1/331)*(25 + 75*alpha - 6*alpha^2 - 5*alpha^3)*alpha^(-1-n).

a(n) = Sum_{i=0..n} Sum_{k=0..n/2} binomial(i+2*k, 2*k)*binomial(i+k, n-i-2*k). - Emanuele Munarini, May 10 2011

Extensions

More terms from James Sellers, Feb 06 2000