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

0,3

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).

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

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 *)

Prog

(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
(Magma) R<x>:=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
(Sage)
def A052960_list(prec):
    P.<x> = 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
(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

Crossrefs

Sequence in context: A114713 A078058 A116413 * A059512 A094297 A026107

Adjacent sequences: A052957 A052958 A052959 * A052961 A052962 A052963

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Feb 06 2000

Status

approved