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A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-3*k-1,n-3*k).

%I A371770 #22 Feb 28 2025 12:07:22
%S A371770 1,2,10,57,338,2057,12741,79914,505954,3226638,20696685,133382658,
%T A371770 862978221,5601919325,36467212610,237974911737,1556281907586,
%U A371770 10196788555859,66921360130374,439860632463462,2895002186799453,19077000179746293,125849150650146714
%N A371770 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-3*k-1,n-3*k).
%H A371770 Robert Israel, <a href="/A371770/b371770.txt">Table of n, a(n) for n = 0..1202</a>
%F A371770 a(n) = [x^n] 1/((1-x^3) * (1-x)^(2*n)).
%F A371770 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
%F A371770 From _Vaclav Kotesovec_, Apr 08 2024: (Start)
%F A371770 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).
%F A371770 a(n) ~ 3^(3*n + 5/2) / (13 * sqrt(Pi*n) * 2^(2*n+1)). (End)
%p A371770 f:= proc(n) local k; add(binomial(3*n-3*k-1,n-3*k),k=0..n/3) end proc:
%p A371770 map(f, [$0..30]); # _Robert Israel_, Feb 28 2025
%o A371770 (PARI) a(n) = sum(k=0, n\3, binomial(3*n-3*k-1, n-3*k));
%Y A371770 Cf. A005809, A165817, A183160.
%Y A371770 Cf. A371758, A371771, A371772.
%K A371770 nonn
%O A371770 0,2
%A A371770 _Seiichi Manyama_, Apr 05 2024