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A371770
a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-3*k-1,n-3*k).
5
1, 2, 10, 57, 338, 2057, 12741, 79914, 505954, 3226638, 20696685, 133382658, 862978221, 5601919325, 36467212610, 237974911737, 1556281907586, 10196788555859, 66921360130374, 439860632463462, 2895002186799453, 19077000179746293, 125849150650146714
OFFSET
0,2
FORMULA
a(n) = [x^n] 1/((1-x^3) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/3-n, 2/3-n, 1-n], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 18*n*(2*n - 1)*(13*n - 22)*(37*n - 51)*a(n) = 3*(40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-1) - (40885*n^4 - 165468*n^3 + 229373*n^2 - 125562*n + 22680)*a(n-2) + 3*(3*n - 5)*(3*n - 4)*(13*n - 9)*(37*n - 14)*a(n-3).
a(n) ~ 3^(3*n + 5/2) / (13 * sqrt(Pi*n) * 2^(2*n+1)). (End)
PROG
(PARI) a(n) = sum(k=0, n\3, binomial(3*n-3*k-1, n-3*k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Apr 05 2024
STATUS
approved