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A100069 Sum(k=0..floor(n/2) binomial(n,k)*4^(n-2*k) ). 3
1, 4, 18, 76, 326, 1384, 5892, 25036, 106438, 452344, 1922588, 8170936, 34726940, 147589264, 627256088, 2665837516, 11329815878, 48151714264, 204644809932, 869740430056, 3696396920116, 15709686864304, 66766169526008, 283756220309176, 1205963937666076, 5125346734404784 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

An inverse Chebyshev transform of x/(1-4x), 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.: x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2)). - corrected by Vaclav Kotesovec, Dec 06 2012

a(n)=sum{k=0..floor(n/2), binomial(n, k)4^(n-2k)}; a(n)=sum{k=0..n, binomial(n, (n-k)/2)(1+(-1)^(n-k)4^k/2}.

Conjecture: +8*n*a(n) +2*(-19*n+4)*a(n-1) +(-15*n-2)*a(n-2) +8*(19*n-23)*a(n-3) +68*(-n+3)*a(n-4)=0. - R. J. Mathar, Nov 22 2012

a(n) ~ 17^n/4^n. - Vaclav Kotesovec, Dec 06 2012

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A027306, A100067, A100068.

Sequence in context: A291417 A017958 A017959 * A058870 A219436 A219137

Adjacent sequences:  A100066 A100067 A100068 * A100070 A100071 A100072

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Nov 02 2004

STATUS

approved

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Last modified July 9 16:50 EDT 2020. Contains 335545 sequences. (Running on oeis4.)