OFFSET
0,2
COMMENTS
Series reversion of x/(1+2x+3x^2). Binomial transform is A107264. Counts colored Motzkin paths. Second binomial transform of 1,0,3,0,18,0,... or 3^n*binomial(n) (A005159) with interpolated zeros.
Hankel transform is 3^binomial(n+1,2). - Paul Barry, Oct 01 2009
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
FORMULA
E.g.f.: exp(2*x)*Bessel_I(1, sqrt(3)*2*x)/(sqrt(3)x).
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*binomial(k)3^k*2^(n-2k).
G.f.: 1/(1-2x-3x^2/(1-2x-3x^2/(1-2x-3x^2/(1-2x-3x^2/(1-.... (continued fraction). - Paul Barry, Oct 01 2009
D-finite with recurrence: (n+2)*a(n) - 2*(2n+1)*a(n-1) + 8*(1-n)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 2*sqrt(9+5*sqrt(3))*(2+2*sqrt(3))^n/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012
MATHEMATICA
CoefficientList[Series[(1-2*x-Sqrt[1-4*x-8*x^2])/(6*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)
PROG
(Sage)
def A122871_list(n): # n>=1
T = [0]*(n+1); R = [1]
for m in (1..n-1):
a, b, c = 1, 0, 0
for k in range(m, -1, -1):
r = a + 2*b + 3*c
if k < m : T[k+2] = u;
a, b, c = T[k-1], a, b
u = r
T[1] = u; R.append(u)
return R
A122871_list(23) # Peter Luschny, Nov 01 2012
(PARI) x='x+O('x^50); Vec((1 - 2*x - sqrt(1 - 4*x - 8*x^2))/(6*x^2)) \\ G. C. Greubel, Mar 19 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 16 2006
STATUS
approved