A101596 G.f.: c(2*x)^4, where c(x) is the g.f. of A000108.

1, 8, 56, 384, 2640, 18304, 128128, 905216, 6449664, 46305280, 334721024, 2434334720, 17801072640, 130809692160, 965500108800, 7154863964160, 53214300733440, 397094950010880, 2972195534929920, 22308469918924800

Offset: 0

Author

Paul Barry, Dec 08 2004

Keywords

• easy
• nonn

Comments

a(n) is also the number of paths in a binary tree of length 2n+3 between two vertices that are 3 steps apart. - David Koslicki, (koslicki(AT)math.psu.edu), Nov 02 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A085687, A003645, A052701.

Programs

Mathematica

CoefficientList[Series[(1-Sqrt[1-8z]+4z(-2+Sqrt[1-8z]+2z))/(32z^4), {z, 0, 20}],z] (* Benedict W. J. Irwin, Jul 12 2016 *)

PARI

x='x+O('x^50); Vec((1-sqrt(1-8*x) + 4*x*(2*x-2+ sqrt(1-8*x)) )/(32*x^4)) \\ G. C. Greubel, May 24 2017

Formula

a(n) = ((8*n+12)/(3*n+12))*((3*n+3)/(n+3))*2^n*C(n+1), where C(n) and the Catalan numbers of A000108.

Conjecture: (n+4)*a(n)-4*(3n+7)*a(n-1)+16*(2n+1)*a(n-2)=0. - R. J. Mathar, Dec 13 2011

From Benedict W. J. Irwin, Jul 12 2016: (Start)

G.f.: (1-sqrt(1-8*x)+4*x*(2*x-2+sqrt(1-8*x)))/(32*x^4).

E.g.f: E^(4*x)*(2*x*(4*x-3)*BesselI(0,4*x) + (3-4*x+ 8*x^2)* BesselI(1, 4*x))/(4*x^3). (End)

a(n) ~ 2^(3*n+5)*n^(-3/2)/sqrt(Pi). - Ilya Gutkovskiy, Jul 12 2016