A155862 A 'Morgan Voyce' transform of A007854.

1, 4, 22, 130, 790, 4870, 30274, 189202, 1186702, 7461982, 47007034, 296527162, 1872479350, 11833642006, 74833075570, 473463268642, 2996771766046, 18974162475598, 120167557286314, 761214481604554, 4822871486667526, 30561172252753030, 193682023673424226, 1227594333811376050, 7781431761074125486

Offset: 0

Author

Paul Barry, Jan 29 2009

Keywords

• easy
• nonn

Comments

Hankel transform is 3^n*2^binomial(n+1, 2).

Image of A007854 by Riordan array (1/(1-x), x/(1-x)^2).

Links

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

Crossrefs

Cf. A001850, A007854, A085478.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(3*Sqrt(1-6*x+x^2) +x -1) )); // G. C. Greubel, Jun 04 2021

Mathematica

CoefficientList[Series[2/(3*Sqrt[1-6*x+x^2]+x-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

Sage

def A155862_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(3*sqrt(1-6*x+x^2) +x-1) ).list()
A155862_list(30) # G. C. Greubel, Jun 04 2021

Formula

G.f.: 2/(3*sqrt(1-6*x+x^2) + x - 1).

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

a(n) = Sum_{k=0..n} binomial(n+k, 2*k)*A007854(k) = Sum_{k=0..n} A085478(n,k) * A007854(k).

2*n*a(n) +(18-25*n)*a(n-1) + 41*(2*n-3)*a(n-2) +(57-25*n)*a(n-3) +2*(n-3)*a(n-4) =0. - R. J. Mathar, Nov 14 2011

a(n) ~ (1+3/sqrt(17)) * (13+3*sqrt(17))^n / 2^(2*n+2). - Vaclav Kotesovec, Feb 01 2014