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Number of partitions of n such that 3*(greatest part) >= (number of parts).
2

%I #15 May 22 2024 15:12:47

%S 1,2,3,4,6,10,14,20,27,38,51,70,92,123,162,212,274,355,453,579,733,

%T 928,1165,1463,1822,2269,2808,3470,4266,5241,6407,7823,9514,11554,

%U 13983,16900,20359,24494,29386,35205,42069,50206,59773,71069,84322,99913,118157,139556,164528,193734

%N Number of partitions of n such that 3*(greatest part) >= (number of parts).

%C Also, the number of partitions of n such that (greatest part) <= 3*(number of parts).

%F G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(3*k+j-1))/(1-x^j).

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(3*k+j-1))/(1-x^j))))

%Y Cf. A064174, A237755, A347868, A347869.

%Y Cf. A237756.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Jan 25 2022