OFFSET
6,2
COMMENTS
It is related to paired pattern P_3 in Section 3.3 in Pan and Remmel's link.
LINKS
G. C. Greubel, Table of n, a(n) for n = 6..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016-2017.
FORMULA
G.f.: -2*(2*x + f(x) - 1)^3/(-2*x + f(x) + 1)^4, where f(x) = sqrt(1 - 4*x).
From Karol A. Penson, Nov 19 2016: (Start)
a(n) = 14*binomial(2*n+1, n-6)/(n+8).
G.f.: 4^7*z^6/(1+sqrt(1-4*z))^14. - shifted by Georg Fischer, Feb 13 2020
E.g.f.: hypergeom([7,15/2],[1,15],4*z). (End)
Recurrence: 2*(n+1)*(2*n+3)*a(n) - (n-5)*(n+9)*a(n+1) = 0. - Georg Fischer, Feb 13 2020
From Amiram Eldar, Nov 04 2025: (Start)
Sum_{n>=6} 1/a(n) = 118*Pi/(63*sqrt(3)) - 74897/32340.
Sum_{n>=6} (-1)^n/a(n) = 745284*log(phi)/(175*sqrt(5)) - 49349197/53900, where phi is the golden ratio (A001622). (End)
a(n) ~ 7 * 4^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 04 2025
MAPLE
a:= n-> 14*binomial(2*n+1, n-6)/(n+8): seq(a(n), n=6..23);
MATHEMATICA
Rest[Rest[Rest[Rest[Rest[Rest[CoefficientList[Series[-((2 (2 x + Sqrt[1 - 4 x] - 1)^3) / (-2 x + Sqrt[1 - 4 x] + 1)^4), {x, 0, 33}], x]]]]]]] (* Vincenzo Librandi, Feb 06 2016 *) (* or *)
RecurrenceTable[{2*(n+1)*(2*n+3)*a[n]-(n-5)*(n+9)*a[n+1]==0, a[6]==1}, a, {n, 6, 25}] (* Georg Fischer, Feb 13 2020 *)
PROG
(PARI) for(n=6, 25, print1(14*binomial(2*n+1, n-6)/(n+8), ", ")) \\ G. C. Greubel, Apr 09 2017, shifted by Georg Fischer, Feb 13 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ran Pan, Feb 04 2016
STATUS
approved