A100069 - OEIS

%I #23 Jun 09 2022 02:25:05

%S 1,4,18,76,326,1384,5892,25036,106438,452344,1922588,8170936,34726940,

%T 147589264,627256088,2665837516,11329815878,48151714264,204644809932,

%U 869740430056,3696396920116,15709686864304,66766169526008,283756220309176,1205963937666076,5125346734404784

%N a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*4^(n-2*k).

%C An inverse Chebyshev transform of x/(1-4*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

%C Generally (for r>1), a(n) ~ (r + 1/r)^n. - _Vaclav Kotesovec_, Dec 06 2012

%H Vincenzo Librandi, <a href="/A100069/b100069.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: x/(sqrt(1-4*x^2)*(2*sqrt(1-4*x^2)+x-2)). - corrected by _Vaclav Kotesovec_, Dec 06 2012

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*4^(n-2*k).

%F a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1 + (-1)^(n-k))*4^k/2.

%F 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

%F a(n) ~ 17^n/4^n. - _Vaclav Kotesovec_, Dec 06 2012

%t 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 *)

%o (PARI) my(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

%o (Magma) m:=4; [(&+[Binomial(n,k)*m^(n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // _G. C. Greubel_, Jun 08 2022

%o (SageMath) m=4; [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

%Y Cf. A027306, A100067, A100068.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Nov 02 2004