A026855 a(n) = T(2n+1,n+2), T given by A026736.

1, 5, 21, 85, 342, 1380, 5598, 22836, 93640, 385734, 1595232, 6619374, 27545269, 114901685, 480282369, 2011058681, 8433331523, 35410037683, 148842787215, 626234799703, 2636930617597, 11111302351505, 46848507630321, 197631791675365

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..1580

Crossrefs

Cf. A000108, A026736.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)*(1-Sqrt(1-4*x))^4/(16*x^3*(Sqrt(1-4*x) -x)) )); // G. C. Greubel, Jul 17 2019

Maple

gf := ((-2*x^3+12*x^2-7*x+1)*sqrt(1-4*x)+16*x^3-24*x^2+9*x-1)/(2*(x^2+4*x-1)*x^3):
S:= series(gf,x,40):
seq(coeff(S,x,j),j=1..30); # Robert Israel, Jan 17 2016

Mathematica

CoefficientList[Series[(1-Sqrt[1-4x])^4/(16*x^4*(1-x/Sqrt[1-4x])), {x, 0, 30}], x] (* David Callan, Jan 16 2016 *)

PARI

my(x='x+O('x^30)); Vec( sqrt(1-4*x)*(1-sqrt(1-4*x))^4/(16*x^3*(sqrt(1-4*x) -x)) ) \\ G. C. Greubel, Jul 17 2019

Sage

a=(sqrt(1-4*x)*(1-sqrt(1-4*x))^4/(16*x^3*(sqrt(1-4*x) -x))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 17 2019

Formula

G.f.: (x*C(x)^4)/(1 - x/sqrt(1 - 4*x)) where C(x) is the g.f. for Catalan numbers A000108. - David Callan, Jan 16 2016

a(n) ~ (3 - sqrt(5))^4 * (2 + sqrt(5))^(n+2) / (16*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019

D-finite with recurrence -2*(n+3)*(26*n-53)*a(n) +(627*n^2-491*n-2440)*a(n-1) +2*(-1234*n^2+3198*n+337)*a(n-2) +(2957*n^2-13637*n+15888)*a(n-3) +2*(211*n-368)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Nov 22 2024