OFFSET
0,8
COMMENTS
FORMULA
a(n) = Sum_{k >= 1} binomial(2*(n+1), (n+1) + 7*k) - 4*binomial(2*n, n+7*k).
From Alois P. Heinz, Jan 21 2023: (Start)
G.f.: (1-sqrt(1-4*x))/(2*x) - (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3).
a(n) = Sum_{k=6..n} A080936(n,k). (End)
D-finite with recurrence -(n+1)*(n-6)*a(n) +3*(3*n^2-17*n+4)*a(n-1) +2*(-13*n^2+80*n-87)*a(n-2) +(25*n^2-161*n+246)*a(n-3) -2*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jan 25 2023
EXAMPLE
a(n) = 0 for n <= 5 because no path of length <= 10 can reach 6 and then descend to 0.
a(6) = 1 because there is one path of length 12 that reaches 6: six steps up, and six steps back down.
MAPLE
a:= n-> binomial(2*n, n)/(n+1)-(<<0|1|0>,
<0|0|1>, <1|-6|5>>^n. <<1, 1, 2>>)[1, 1]:
seq(a(n), n=0..35); # Alois P. Heinz, Jan 21 2023
MATHEMATICA
Table[Sum[Binomial[2(n + 1), (n + 1) + 7 k] - 4 Binomial[2n, n + 7k], {k, 1, n}], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Greg Dresden, Jan 21 2023
STATUS
approved