OFFSET
0,3
FORMULA
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n,k) * binomial(3*n+k+1,n) / (3*n+k+1).
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n,k) * binomial(4*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n,k) * binomial(3*n,k-1) for n > 0.
D-finite with recurrence +270*n*(3*n-1)*(3*n+1)*a(n) +(-9463*n^3 -45948*n^2 +88297*n -35478)*a(n-1) +36*(-9017*n^3 +49691*n^2 -90408*n +54354)*a(n-2) +48*(53*n^3 +1724*n^2 -11161*n +16518)*a(n-3) +576*(3*n-10)*(3*n-11) *(n-4)*a(n-4)=0. - R. J. Mathar, Aug 16 2023
MAPLE
A364923 := proc(n)
add( 3^k*(-2)^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1), k=0..n) ;
end proc:
seq(A364923(n), n=0..80); # R. J. Mathar, Aug 16 2023
PROG
(PARI) a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(3*n+k+1, n)/(3*n+k+1));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 12 2023
STATUS
approved