%I #21 May 30 2024 06:55:31

%S 0,0,0,1,1,2,3,5,5,8,9,13,14,18,20,27,28,35,38,49,51,61,66,81,86,102,

%T 109,130,136,161,172,202,214,245,264,305,323,369,395,452,480,544,580,

%U 657,703,786,842,947,1008,1124,1205,1340,1432,1589,1702,1886,2014

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

%H David A. Corneth, <a href="/A237825/b237825.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} x^(4*k)/Product_{j=k..3*k} (1-x^j). - _Seiichi Manyama_, May 14 2023

%e a(7) = 3 counts these partitions: 331, 3211, 31111.

%t z = 64; q[n_] := q[n] = IntegerPartitions[n];

%t Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}] (* A237825*)

%t Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}] (* A237826 *)

%t Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}] (* A237827 *)

%t Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)

%t Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)

%t Table[Count[IntegerPartitions[n],_?(3#[[-1]]==#[[1]]&)],{n,60}] (* _Harvey P. Dale_, May 14 2023 *)

%t kmax = 57;

%t Sum[x^(4 k)/Product[1 - x^j, {j, k, 3 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* _Jean-François Alcover_, May 30 2024, after _Seiichi Manyama_ *)

%o (PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, 3*k, 1-x^j)))) \\ _Seiichi Manyama_, May 14 2023

%Y Cf. A000041, A117086, A237757, A237828, A237829.

%Y Cf. A118096, A237826, A237827.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Feb 16 2014