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

0,2

Comments

Hankel transform is A165410.

Links

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

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=0..n} if(n<=3k, 2^k*C((n+k)/2, k)*((3*k-n)/2 + 1)(1+(-1)^(n-k))/(2*(k+1)) = 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

Mathematica

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

Prog

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

Crossrefs

Cf. A000108, A000129, A165408, A165410.

Sequence in context: A052912 A024740 A025275 * A052542 A163271 A178036

Adjacent sequences: A165406 A165407 A165408 * A165410 A165411 A165412

Keyword

easy,nonn

Author

Paul Barry, Sep 17 2009

Status

approved