OFFSET
3,2
COMMENTS
This sequence is related to paired pattern P_2 in Pan and Remmel's link.
LINKS
G. C. Greubel, Table of n, a(n) for n = 3..1000
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
FORMULA
G.f.: -((-1 + f(x))^3*x*(-1 + f(x) + 2*x))/(2*(1 - f(x) + (-5 + f(x))*x)^3), where f(x) = sqrt(1 - 4*x).
a(n) = ((Sum_{m=1..n-3}((m+2)*(Sum_{k=0..m/2}((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k,k)))*binomial(2*n-m-5,n-m-3)))+2*binomial(2*n-4,n-2))/(2*n-2), n>2. - Vladimir Kruchinin, Feb 28 2016
a(n) ~ 7*2^(2*n+2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 28 2016
D-finite with recurrence (n-1)*(9*n-56)*a(n) +(-31*n^2+203*n-16)*a(n-1) +(-193*n^2+1371*n-2138)*a(n-2) +3*(217*n^2-1497*n+2274)*a(n-3) +2*(41*n-94)*(2*n-5)*a(n-4)=0. - R. J. Mathar, Jul 24 2022
MATHEMATICA
Rest[Rest[Rest[CoefficientList[Series[-((-1 + Sqrt[1 - 4 x])^3 x (-1 + Sqrt[1-4 x] + 2 x))/(2 (1 - Sqrt[1 - 4 x] + (-5 + Sqrt[1 - 4 x]) x)^3), {x, 0, 40}], x]]]] (* Vincenzo Librandi, Feb 28 2016 *)
PROG
(Maxima)
a(n):=((sum((m+2)*(sum((m^2+(3-2*k)*m+k^2-3*k+2)*binomial(m-k, k), k, 0, m/2)) *binomial(2*n-m-5, n-m-3), m, 1, n-3))+2*binomial(2*n-4, n-2))/(2*n-2); /* Vladimir Kruchinin, Feb 28 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Ran Pan, Feb 03 2016
STATUS
approved