A237827 - OEIS

%I #26 May 30 2024 21:58:33

%S 0,0,0,0,0,1,1,2,3,5,7,11,13,19,24,32,39,52,61,77,93,114,133,164,188,

%T 226,261,309,353,417,471,549,622,717,808,933,1042,1191,1334,1516,1690,

%U 1921,2131,2407,2674,3006,3330,3744,4135,4628,5116,5708,6294,7020

%N Number of partitions of n such that 5*(least part) = greatest part.

%H David A. Corneth, <a href="/A237827/b237827.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Alois P. Heinz)

%F G.f.: Sum_{k>=1} x^(6*k)/Product_{j=k..5*k} (1-x^j). - _Seiichi Manyama_, May 14 2023

%e a(8) = 2 counts these partitions:  521, 5111.

%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 (* Second program: *)

%t kmax = 54;

%t Sum[x^(6 k)/Product[1 - x^j, {j, k, 5 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, 0, 0], Vec(sum(k=1, N, x^(6*k)/prod(j=k, 5*k, 1-x^j)))) \\ _Seiichi Manyama_, May 14 2023

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

%Y Cf. A118096, A237825, A237826.

%K nonn,easy

%O 1,8

%A _Clark Kimberling_, Feb 16 2014