OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Robert Israel, Recurrence of order 9
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+1,k) * binomial(4*k,n-k).
G.f.: (1/x) * Series_Reversion( x / (1 + x * (1 + x)^4) ). - Seiichi Manyama, Oct 03 2025
D-finite with recurrence of order 9 (see link). - Robert Israel, Aug 03 2026
MAPLE
with(gfun):
eq:= (y*(1-x*(1+x*y)^4)-1):
de:= algeqtodiffeq(eq, y(x)):
rec:= diffeqtorec(de, y(x), a(n)):
f:= rectoproc(rec, a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 03 2026
PROG
(PARI) a(n) = sum(k=0, n, binomial(n+1, k)*binomial(4*k, n-k))/(n+1);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 05 2023
STATUS
approved
