OFFSET
0,2
LINKS
Robert Israel, Table of n, a(n) for n = 0..865
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..n} 25^k * binomial(n/5+1/5,k) * binomial(k,n-k).
a(5*n+4) = 0 for n >= 0.
a(n) = 25^n*binomial((n+1)/5, n)*hypergeom([(1-n)/2, -n/2], [(6-4*n)/5], 4/25)/(n+1). - Stefano Spezia, Apr 18 2024
D-finite with recurrence: -1344056888671875*(3*n + 8)*(n + 6)*(3*n - 2)*(n + 1)*a(n) - 723515625*(n + 6)*(5762434*n^3 + 60414477*n^2 + 201661457*n + 212482114)*a(n + 5) + 39375*(33164403*n^4 + 1373285372*n^3 + 21229796013*n^2 + 145178949672*n + 370474731340)*a(n + 10) - 300*(15234*n^4 + 889091*n^3 + 19781899*n^2 + 198829036*n + 760953740)*a(n + 15) + 4*(n + 20)*(n + 19)*(n + 18)*(n + 17)*a(n + 20) = 0. - Robert Israel, Aug 07 2026
MAPLE
with(gfun):
eq := y^5 - 1 - 25*x*y*(x*y + 1):
de:= subs(RootOf(_Z^5-1)=1, algeqtodiffeq(eq, y(x))):
rec:= diffeqtorec(de, y(x), a(n)):
f:= rectoproc(rec, a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 07 2026
PROG
(PARI) a(n) = sum(k=0, n, 25^k*binomial(n/5+1/5, k)*binomial(k, n-k))/(n+1);
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Apr 15 2024
STATUS
approved