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A100068 Sum(k=0..floor(n/2), binomial(n,k)*3^(n-2*k) ). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

An inverse Chebyshev transform of x/(1-3x), 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(1-4x^2))A(xc(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-2k)} has g.f. 2x/(sqrt(1-4x^2)(r*sqrt(1-4x^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

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}.

Conjecture: 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

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 *)

PROG

(PARI) 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

CROSSREFS

Cf. A027306, A100067, A100069.

Sequence in context: A017937 A017938 A333760 * A260745 A119213 A068644

Adjacent sequences:  A100065 A100066 A100067 * A100069 A100070 A100071

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 02 2004

STATUS

approved

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Last modified July 13 13:48 EDT 2020. Contains 335688 sequences. (Running on oeis4.)