OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
FORMULA
a(n) = Sum_{i=0,..,n} (-1)^i*(i+1)*binomial(3*n-2*i, n-i)/(2*n-i+1).
G.f.: g/(1+zg) where g = 1 + z*g^3, g(0) = 1.
G.f.: g/(1+zg) where g = 2*sin(arcsin(sqrt(27*z)/2)/3)/sqrt(3*z).
G.f.: A(x)=1/(1+x)*(1+6*x/((1+x)*G(0)-6*x)) ; G(k)= 3*x*(3*k+1)*(3*k+2) + (2*k+2)*(2*k+3) - 6*x*(k+1)*(2*k+3)*(3*k+4)*(3*k+5)/G(k+1) ; (continued fraction Euler's kind,1-step ). - Sergei N. Gladkovskii, Dec 29 2011
a(n) ~ 27^(n+3/2) / (121 * sqrt(Pi) * 4^(n+1) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence 2*n*(2*n+1)*a(n) +(-47*n^2+65*n-24)*a(n-1) +3*(49*n^2-167*n+148)*a(n-2) +(-65*n^2+365*n-396)*a(n-3) -12*(3*n-5)*(3*n-7)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
MATHEMATICA
Table[Sum[(-1)^i*(i+1)*Binomial[3*n-2*i, n-i]/(2*n-i+1), {i, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 17 2014 *)
PROG
(PARI) for(n=0, 50, print1(sum(k=0, n, (-1)^k*(k+1)*binomial(3*n-2*k, n-k)/(2*n - k+1)), ", ")) \\ G. C. Greubel, Feb 07 2017
(PARI) Vec((g->g/(1+x*g))(1 + serreverse(x/(1+x)^3 + O(x^25)))) \\ Andrew Howroyd, Nov 12 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Offset corrected by Vaclav Kotesovec, Mar 17 2014
STATUS
approved