A126932 Binomial transform of A127358.

1, 4, 15, 55, 199, 714, 2547, 9048, 32043, 113212, 399265, 1406079, 4946137, 17383162, 61048359, 214270215, 751691811, 2636004228, 9240836733, 32386215981, 113478349989, 397544907486, 1392493797765, 4876916883090, 17078574481941, 59802541979964

Offset: 0

Author

Philippe Deléham, Mar 17 2007

Keywords

• nonn

Comments

Hankel transform is (-1)^n.

Row sums of the Riordan array ((1-2*x)/(1+x+x^2), x/(1+x+x^2))^(-1). - Paul Barry, Nov 06 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)) )); // G. C. Greubel, Jan 29 2020

Maple

seq(coeff(series( (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Jan 29 2020

Mathematica

CoefficientList[Series[(Sqrt[-3*x^2-2*x+1]-3*(3*x-1))/(2*(21*x^2-13*x+2)), {x, 0, 30}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

PARI

my(x='x+O('x^30)); Vec( (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)) ) \\ G. C. Greubel, Jan 29 2020

Sage

def A126932_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)) ).list()
A126932_list(30) # G. C. Greubel, Jan 29 2020

Formula

a(n+1) = 3*a(n) + A059738(n) with a(0)=1.

G.f: (sqrt(1-2*x-3*x^2) + 3*(1-3*x))/(2*(2-13*x+21*x^2)). - Paul Barry, Nov 06 2008

Conjecture: +2*n*a(n) -11*n*a(n-1) +4*(2*n+3)*a(n-2) +21*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ 3 * 7^n / 2^(n+1). - Vaclav Kotesovec, Feb 12 2014

Extensions

Corrected and extended by Vincenzo Librandi, Feb 13 2014