OFFSET
8,2
COMMENTS
It is related to paired pattern P_3 in Section 3.3 in Pan and Remmel's link.
LINKS
Robert Israel, Table of n, a(n) for n = 8..1658
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
FORMULA
G.f.: (2*(-1 + f(x) + 2*x)^4)/(1 + f(x) - 2*x)^5, where f(x) = sqrt(1 - 4*x).
Conjecture: -(n+10)*(n-8)*a(n) +2*n*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jun 07 2016
Conjecture follows from the differential equation (6*x + 80)*y(x) + (14*x^2 - 3*x)*y'(x) + (4*x^3 - x^2)*y''(x) = 0 satisfied by the G.f. - Robert Israel, May 15 2025
a(n) = 18*(2*n+1)!/((n-8)!*(n+10)!). - Vaclav Kotesovec, May 15 2025
E.g.f.: 9*exp(2*x)*BesselI(9,2*x)/x. - Stefano Spezia, Nov 24 2025
MAPLE
f:= proc(n) option remember;
2*n*(2*n+1)*procname(n-1)/(10+n)/(n-8)
end proc:
f(8):= 1:
map(f, [$8 .. 40]); # Robert Israel, May 15 2025
MATHEMATICA
Rest[Rest[Rest[Rest[Rest[Rest[Rest[Rest[CoefficientList[Series[(2 (-1 + Sqrt[1 - 4 x] + 2 x)^4) / (1 + Sqrt[1 - 4 x] - 2 x)^5, {x, 0, 33}], x]]]]]]]]] (* Vincenzo Librandi, Feb 06 2016 *)
Table[18*(2*n + 1)!/((n - 8)!*(n + 10)!), {n, 8, 30}] (* Vaclav Kotesovec, May 15 2025 *)
PROG
(PARI) x='x+O('x^100); Vec((2*(-1 + (1 - 4*x)^(1/2) + 2*x)^4)/(1 + (1 - 4*x)^(1/2) - 2*x)^5) \\ Altug Alkan, Feb 04 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Ran Pan, Feb 04 2016
STATUS
approved