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A101596 G.f.: c(2*x)^4, where c(x) is the g.f. of A000108. 1
1, 8, 56, 384, 2640, 18304, 128128, 905216, 6449664, 46305280, 334721024, 2434334720, 17801072640, 130809692160, 965500108800, 7154863964160, 53214300733440, 397094950010880, 2972195534929920, 22308469918924800 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A199939 A003494 A057084 * A327834 A092521 A156088
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 08 2004
STATUS
approved

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)