OFFSET
0,2
COMMENTS
Binomial transform of A126932. Hankel transform is (-1)^n.
Row sums of the Riordan matrix (1/(1-4*x),(1-sqrt(1-4*x))/(2*sqrt(1-4*x)) (A188481). - Emanuele Munarini, Apr 01 2001
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
FORMULA
a(n) = Sum_{k=0..n} C(2n-k,n-k)*3^k.
From Emanuele Munarini, Apr 01 2011: (Start)
a(n) = [x^n] 1/((1+x)^(n+1)*(1-3x)).
a(n) = 3^(2n+1)/2^(n+2) + (1/4)*sum(binomial(2k,k)*(9/2)^(n-k),k=0..n).
D-finite with recurrence: 2*(n+2)*a(n+2) - (17*n+30)*a(n+1) + 18*(2*n+3)*a(n) = 0.
G.f.: (3-12*x+sqrt(1-4*x))/(4-34*x+72*x^2). (End)
G.f.: (1/(1-4*x)^(1/2)+3)/(4-18*x)=( 2 + x/(Q(0)-2*x))/(2-9*x) where Q(k) = 2*(2*k+1)*x + (k+1) - 2*(k+1)*(2*k+3)*x/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Mar 18 2013
a(n) ~ 3^(2*n + 1) / 2^(n + 1). - Vaclav Kotesovec, Sep 15 2021
MATHEMATICA
CoefficientList[Series[(3-12x+Sqrt[1-4x])/(4-34x+72x^2), {x, 0, 100}], x] (* Emanuele Munarini, Apr 01 2011 *)
PROG
(Maxima) makelist(sum(binomial(n+k, k)*3^(n-k), k, 0, n), n, 0, 12); /* Emanuele Munarini, Apr 01 2011 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jun 14 2008
STATUS
approved