OFFSET
0,5
LINKS
Robert Israel, Table of n, a(n) for n = 0..2271
FORMULA
a(n) ~ 2^(5/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025
Conjecture D-finite with recurrence +3*(n+4)*(3*n+4)*(3*n+8)*a(n) +3*(-63*n^3-297*n^2-349*n-60)*a(n-1) +3*(189*n^3+270*n^2-229*n-140)*a(n-2) +15*(-63*n^3+117*n^2+44*n-64)*a(n-3) +(689*n^3-5372*n^2+6946*n-1288)*a(n-4) +(n-4)*(201*n^2+2767*n-3011)*a(n-5) -(n-5)*(579*n+257)*(n-4)*a(n-6) +229*(n-5)*(n-6)*(n-4)*a(n-7)=0. - R. J. Mathar, Jan 29 2025
EXAMPLE
For n = 4, the a(4)=4 paths are HHHH, UDUU, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).
MAPLE
f:= proc(n, y) option remember;
if n = 0 then if y = 0 then return 1 else return 0 fi fi;
if y > n then return 0 fi;
if y >= -1 then procname(n-1, y-1) + procname(n-1, y) + procname(n-1, y+3)
else procname(n-1, y) + procname(n-1, y+3)
fi;
end proc:
map(f, [$0..40], 0); # Robert Israel, Jan 23 2025
PROG
(PARI) a(n) = sum(k=0, floor(n/4), 3*binomial(n, k*4)*binomial(4*k+3, k)/(4*k+3)) \\ Thomas Scheuerle, Jan 07 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Emely Hanna Li Lobnig, Dec 23 2024
EXTENSIONS
More terms from Jinyuan Wang, Jan 07 2025
STATUS
approved