A394189 - OEIS

%I #20 Mar 13 2026 09:58:20

%S 1,0,1,1,2,1,3,1,3,3,4,3,8,5,7,11,11,10,17,14,19,22,21,22,35,28,33,41,

%T 45,42,62,52,68,76,80,81,117,98,121,136,156,147,196,178,224,236,257,

%U 255,340,304,363,382,440,422,526,496,599,606,677,670,845,780,910,934,1079,1051,1249

%N Number of partitions of n such that (number of distinct parts) < minimal multiplicity of the parts.

%H Seiichi Manyama, <a href="/A394189/b394189.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Sum_{j>=0} [z^j] Product_{k>=1} (1 + z*q^((j+1)*k)/(1-q^k)).

%e a(12) = 8 counts these partitions: 66, 444, 3333, 333111, 222222, 22221111, 222111111, 111111111111.

%o (PARI) my(N=70, q='q+O('q^N)); Vec(sum(j=0, N, polcoef(prod(k=1, N, 1+z*q^((j+1)*k)/(1-q^k)), j, z)))

%Y Cf. A239966, A240305, A394188.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Mar 12 2026