A026773 a(n) = T(2n-1,n-1), T given by A026769. Also T(2n+1,n+1), T given by A026780.

1, 4, 17, 76, 352, 1674, 8129, 40156, 201236, 1020922, 5234660, 27089726, 141335846, 742712598, 3927908193, 20891799036, 111688381228, 599841215226, 3234957053984, 17512055200470, 95125188934942, 518340392855286, 2832580291316092, 15520177744727766

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A026769, A026770, A026771, A026772, A026774, A026775, A026776, A026777, A026778, A026779.

Programs

GAP

List([0..30], n-> Sum([0..n], k-> Binomial(n+1,k)*Binomial(n+k+1, n+1)/(k+1) )); # G. C. Greubel, Nov 01 2019

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt(1-4*x) - Sqrt(1-6*x+x^2))/(2*x) -1/2 )); // G. C. Greubel, Nov 01 2019

Maple

seq(coeff(series((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2, x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 01 2019

Mathematica

Rest@CoefficientList[Series[(Sqrt[1-4*x] - Sqrt[1-6*x+x^2])/(2*x) -1/2, {x,0,30}], x] (* G. C. Greubel, Nov 01 2019 *)

PARI

my(x='x+O('x^30)); Vec((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2) \\ G. C. Greubel, Nov 01 2019

Sage

def A026773_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2).list()
a=A026773_list(30); a[1:] # G. C. Greubel, Nov 01 2019

Formula

From Vladeta Jovovic, Nov 23 2003: (Start)

a(n) = A006318(n) - A000108(n).

G.f.: (sqrt(1-4*x) - sqrt(1-6*x+x^2))/(2*x) -1/2. (End)

From Paul Barry, May 19 2005: (Start)

a(n) = Sum_{k=0..n} C(n+k+1, n+1)*C(n+1, k)/(k+1).

a(n) = Sum_{k=0..n+1} C(n+2, k)*C(n+k, n+1)/(n+2). (End)

D-finite with recurrence n*(n+1)*a(n) -n*(11*n-7)*a(n-1) +(37*n^2-95*n+54)*a(n-2) +(-49*n^2+269*n-354)*a(n-3) +6*(9*n^2-71*n+138)*a(n-4) -4*(2*n-9)*(n-5)*a(n-5)=0. - R. J. Mathar, Aug 05 2021