OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..972
FORMULA
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(4*n-3*k,n-1-k) for n > 0.
From Seiichi Manyama, Dec 05 2024: (Start)
G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^3/(1 - x*A(x))).
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r). (End)
G.f.: 1 + Series_Reversion( x / ((1+x) * (x+(1+x)^3)) ). - Seiichi Manyama, Sep 28 2025
D-finite with recurrence: -403*n*(n + 2)*(n + 1)*a(n) + 5*(133*n + 246)*(n + 2)*(n + 1)*a(n + 1) - 45*(15*n + 74)*(n + 3)*(n + 2)*a(n + 2) + (4505*n^3 + 37365*n^2 + 105142*n + 101376)*a(n + 3) - (1361*n^3 + 15219*n^2 + 54526*n + 61320)*a(n + 4) + 3*(1489*n^3 + 23017*n^2 + 118192*n + 201564)*a(n + 5) - (697*n^3 + 12579*n^2 + 75680*n + 151812)*a(n + 6) + 3*(3*n + 20)*(3*n + 22)*(n + 7)*a(n + 7) = 0. - Robert Israel, Aug 04 2026
MAPLE
with(gfun):
eq:= (-y+1)*(1-x*y)+x*y^4:
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 04 2026
PROG
(PARI) a(n) = if(n==0, 1, sum(k=0, n-1, binomial(n, k)*binomial(4*n-3*k, n-1-k))/n);
(PARI) a(n, r=1, s=1, t=4, u=1) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 05 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 05 2023
STATUS
approved