A165409 Transform of 2^n by the aerated Catalan triangle A165408.

1, 2, 4, 10, 24, 56, 136, 328, 784, 1896, 4576, 11008, 26592, 64192, 154752, 373696, 902144, 2176640, 5255424, 12687488, 30621952, 73931392, 178484736, 430845952, 1040176640, 2511199232, 6062209024, 14635617280, 35333443584, 85300015104

Offset: 0

Author

Paul Barry, Sep 17 2009

Keywords

• easy
• nonn

Comments

Hankel transform is A165410.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A000129, A165408, A165410.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2/(1-4*x+Sqrt(1-8*x^3)) )); // G. C. Greubel, Nov 10 2022

Mathematica

CoefficientList[Series[2/(1-4*x+Sqrt[1-8*x^3]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

SageMath

def A165408(n,k): return 0 if (n>3*k) else (1+(-1)^(n-k))*(3*k-n+2)*binomial(int((n+k)/2), k)/(4*(k+1))
def A165409(n): return sum(2^k*A165408(n,k) for k in range(n+1))
[A165409(n) for n in range(41)] # G. C. Greubel, Nov 10 2022

Formula

G.f.: 1/(1-2*x-2*x^3*c(2*x^3)) = 2/(1-4*x+sqrt(1-8*x^3)) = (1-4*x-sqrt(1-8*x^3) )/(4*x*(1-2*x-x^2)), c(x) the g.f. of A000108.

G.f.: 1/(1-2*x-2*x^3/(1-2*x^3/(1-2*x^3/(1-2*x^3/(1-... (continued fraction).

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

a(n) = Sum_{k=0..n} 2^k * A165408(n,k).

a(n) = Sum_{k=0..n+1} Pell(n-k+1)*(0^k - 2^((k-2)/2)*A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).

(n+1)*a(n) = 2(n+1)*a(n-1) + (n+1)*a(n-2) + 4*(2*n-7)*a(n-3) - 8(2*n-7)*a(n-4) - 4*(2*n-7)*a(n-5). - R. J. Mathar, Nov 17 2011

a(n) ~ (4+sqrt(2)) * (1+sqrt(2))^n / 8. - Vaclav Kotesovec, Feb 01 2014