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A115962 Expansion of 1/sqrt(1-4*x^2-4*x^3). 3
1, 0, 2, 2, 6, 12, 26, 60, 130, 300, 672, 1540, 3514, 8064, 18552, 42756, 98802, 228624, 530024, 1230372, 2860000, 6655792, 15505932, 36159552, 84398626, 197154984, 460903796, 1078251044, 2524144224, 5912535672, 13857378300, 32495267712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Diagonal sums of number triangle A115951.

Number of lattice paths from (0,0) to (n,n) using steps (2,1), (1,0), (1,2). - Joerg Arndt, Jul 05 2011

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = Sum_{k=0..floor(n/2)} C(2*k,k)*C(k,n-2*k).

G.f.: Q(0), where Q(k) = 1 + 4*x*(x+x^2)*(4*k+1) / (4*k+2 - 4*x*(x+x^2)*(4*k+2)*(4*k+3) / (4*x*(x+x^2)*(4*k+3) + 4*(k+1) / Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013

D-finite: n*a(n) - 4*(n-1)*a(n-2) - 2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020

MAPLE

A115962 := proc(n)

    option remember;

    if n < 4 then

        op(n+1, [1, 0, 2, 2]);

    else

        4*(n-1)*procname(n-2)+2*(2*n-3)*procname(n-3) ;

        %/n ;

    end if;

end proc:

seq(A115962(n), n=0..20) ; # R. J. Mathar, Jan 14 2020

MATHEMATICA

CoefficientList[Series[1/Sqrt[1-4x^2-4x^3], {x, 0, 35}], x] (* or *) Table[Sum[Binomial[2k, k] Binomial[k, n-2k], {k, 0, Floor[n/2]}], {n, 0, 35}] (* Michael De Vlieger, Sep 03 2015 *)

PROG

(PARI) x = xx+O(xx^40); Vec(1/sqrt(1-4*x^2-4*x^3)) \\ Michel Marcus, Sep 03 2015

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1-4*x^2-4*x^3) )); // G. C. Greubel, May 06 2019

(Sage) (1/sqrt(1-4*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 06 2019

CROSSREFS

Sequence in context: A324128 A217211 A035615 * A019311 A216215 A052994

Adjacent sequences:  A115959 A115960 A115961 * A115963 A115964 A115965

KEYWORD

easy,nonn,changed

AUTHOR

Paul Barry, Mar 14 2006

STATUS

approved

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)