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

0,2

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

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

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 *)

Prog

(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

Crossrefs

Cf. A085687, A003645, A052701.

Sequence in context: A199939 A003494 A057084 * A327834 A092521 A156088

Adjacent sequences: A101593 A101594 A101595 * A101597 A101598 A101599

Keyword

easy,nonn

Author

Paul Barry, Dec 08 2004

Status

approved