A099140 a(n) = 4^n * T(n,3/2) where T is the Chebyshev polynomial of the first kind.

1, 6, 56, 576, 6016, 62976, 659456, 6905856, 72318976, 757334016, 7930904576, 83053510656, 869747654656, 9108115685376, 95381425750016, 998847258034176, 10460064284409856, 109539215284371456, 1147109554861899776

Offset: 0

Author

Paul Barry, Sep 30 2004

Keywords

• easy
• nonn

Comments

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2n,2k) and a(n) = (1+sqrt(r+1))^(2n)/2 + (1-sqrt(r+1))^(2n)/2.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..900
• P. J. Szablowski, On moments of Cantor and related distributions, arXiv preprint arXiv:1403.0386 [math.PR], 2014.
• Index entries for linear recurrences with constant coefficients, signature (12,-16).

Crossrefs

Cf. A001541, A081294, A083884, A090965, A099141, A099142.

Programs

Mathematica

LinearRecurrence[{12,-16},{1,6},30] (* Harvey P. Dale, Oct 23 2012 *)

PARI

a(n) = 4^n*polchebyshev(n, 1, 3/2); \\ Michel Marcus, Sep 08 2019

Formula

G.f.: (1-6*x)/(1-12*x+16*x^2);

E.g.f.: exp(6*x)*cosh(2*sqrt(5)*x);

a(n) = 4^n * T(n, 6/4) where T is the Chebyshev polynomial of the first kind;

a(n) = Sum_{k=0..n} 5^k*binomial(2n, 2k);

a(n) = (1+sqrt(5))^(2n)/2 + (1-sqrt(5))^(2n)/2.

a(n) = a(0)=1, a(1)=6, 12*a(n-1) - 16*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009