A394029 - OEIS

%I #15 Mar 08 2026 16:20:50

%S 0,0,0,1,1,2,2,4,3,5,5,7,5,6,6,9,7,10,9,14,14,17,19,24,26,30,34,37,40,

%T 48,52,58,63,68,75,84,91,101,108,120,131,145,154,170,188,208,222,248,

%U 264,294,318,348,378,417,450,490,529,576,622,680,732,795,858,926,997,1080,1160

%N Number of partitions p of n with multiplicity of each part at most 3, satisfying max(p) = 3 * min(p).

%H Vincenzo Librandi, <a href="/A394029/b394029.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: Sum_{j>=1} q^(4*j)*(1-q^(3*j))*(1-q^(9*j))/((1-q^j)*(1-q^(3*j))) * Product_{k=j+1..3*j-1} (1-q^(4*k))/(1-q^k).

%t Nmax=80; Rest@CoefficientList[Series[Sum[q^(4*j)*(1-q^(3*j))*(1-q^(9*j))/((1-q^j)*(1-q^(3*j)))*Product[(1-q^(4*k))/(1-q^k),{k,j+1,3*j-1}],{j,1,Nmax}],{q,0,Nmax}]//Normal,q] (* _Vincenzo Librandi_, Mar 08 2026 *)

%o (Magma) N := 80; R<q> := PowerSeriesRing(Integers(), N+5); gf := &+[ q^(4*j)*(1-q^(3*j))*(1-q^(9*j))/((1-q^j)*(1-q^(3*j))) * &*[(1-q^(4*k))/(1-q^k) : k in [j+1..3*j-1]] : j in [1..N] ]; [Coefficient(gf,n) : n in [1..N]]; // _Vincenzo Librandi_, Mar 08 2026

%Y Cf. A241063, A394024, A394028.

%K nonn

%O 1,6

%A _Seiichi Manyama_, Mar 07 2026