OFFSET
0,2
LINKS
Robert Israel, Table of n, a(n) for n = 0..762
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..n} 9^k * binomial(n/3+k-2/3,k) * binomial(k,n-k).
a(n) = 9^n*binomial((4*n-2)/3, n)*hypergeom([(1-n)/2, -n/2], [(2-4*n)/3], -4/9)/(n+1). - Stefano Spezia, Apr 18 2024
D-finite with recurrence: 139606401420*(5*n + 11)*(5*n + 2)*(5*n + 8)*(5*n + 14)*a(n) + 81*(1516846256131*n^4 + 18335525262694*n^3 + 83947437723741*n^2 + 176041287279394*n + 145874829503920)*a(n + 3) + 351*(25995607591*n^4 + 689078023582*n^3 + 6794758051881*n^2 + 29533754487562*n + 47731677004000)*a(n + 6) - 507*(n + 10)*(20297633*n^3 + 617423604*n^2 + 6299626243*n + 21501443664)*a(n + 9) + 1069939*(n + 10)*(n + 13)*(n + 12)*(n + 11)*a(n + 12) = 0. - Robert Israel, May 21 2026
MAPLE
f:= gfun:-rectoproc({139606401420*(5*n + 11)*(5*n + 2)*(5*n + 8)*(5*n + 14)*a(n) + 81*(1516846256131*n^4 + 18335525262694*n^3 + 83947437723741*n^2 + 176041287279394*n + 145874829503920)*a(n + 3) + 351*(25995607591*n^4 + 689078023582*n^3 + 6794758051881*n^2 + 29533754487562*n + 47731677004000)*a(n + 6) - 507*(n + 10)*(20297633*n^3 + 617423604*n^2 + 6299626243*n + 21501443664)*a(n + 9) + 1069939*(n + 10)*(n + 13)*(n + 12)*(n + 11)*a(n + 12), a(0) = 1, a(1) = 3, a(2) = 30, a(3) = 378, a(4) = 5382, a(5) = 82377, a(6) = 1323153, a(7) = 21998493, a(8) = 375346062, a(9) = 6534966438, a(10) = 115634273139, a(11) = 2073448947960}, a(n), remember):
map(f, [$0..20]); # Robert Israel, May 21 2026
PROG
(PARI) a(n) = sum(k=0, n, 9^k*binomial(n/3+k-2/3, k)*binomial(k, n-k))/(n+1);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 17 2024
STATUS
approved