A100068 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*3^(n-2*k).

1, 3, 11, 36, 123, 408, 1370, 4560, 15235, 50760, 169326, 564336, 1881582, 6271632, 20907156, 69689376, 232304355, 774343560, 2581169510, 8603882160, 28679699578, 95598937008, 318663476076, 1062211351776, 3540705857998, 11802351958608, 39341178395660, 131137257852000

Offset: 0

Author

Paul Barry, Nov 02 2004

Keywords

• easy
• nonn

Comments

An inverse Chebyshev transform of x/(1-3*x), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. In general, Sum_{k=0..floor(n/2)} binomial(n,k) * r^(n-2*k) has g.f. 2*x/(sqrt(1-4*x^2)*(r*sqrt(1-4*x^2) + 2*x - r)). - corrected by Vaclav Kotesovec, Dec 06 2012

Generally (for r>1), a(n) ~ (r + 1/r)^n. - Vaclav Kotesovec, Dec 06 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A027306, A100067, A100069.

Programs

Magma

m:=3; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 08 2022

Mathematica

CoefficientList[Series[2*x/(Sqrt[1-4*x^2]*(3*Sqrt[1-4*x^2] + 2*x-3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 06 2012 *)

PARI

my(x='x+O('x^66)); Vec(2*x/(sqrt(1-4*x^2)*(3*sqrt(1-4*x^2)+2*x-3))) \\ Joerg Arndt, May 12 2013

SageMath

m=3; [sum(binomial(n,k)*m^(n-2*k) for k in (0..n//2)) for n in (0..40)] # G. C. Greubel, Jun 08 2022

Formula

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*3^(n-k).

a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1+(-1)^(n-k))*3^k/2.

D-finite with recurrence 9*n*a(n) +12*(-3*n+1)*a(n-1) +4*(-4*n-1)*a(n-2) +48*(3*n-4)*a(n-3) +80*(-n+3)*a(n-4)=0. - R. J. Mathar, Nov 22 2012

a(n) ~ 10^n/3^n. - Vaclav Kotesovec, Dec 06 2012