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%I #12 May 17 2023 08:34:25
%S 1,1,1,1,2,1,2,3,2,2,5,3,4,5,5,6,8,6,8,10,10,10,15,12,14,17,18,20,23,
%T 21,26,29,30,31,39,38,42,46,49,52,61,60,68,74,77,83,94,95,104,112,122,
%U 128,143,144,159,172,181,192,212,219,237,253,271,285
%N Number of partitions of n such that 2*(least part) - 1 = greatest part.
%F G.f.: x + Sum_{k>=1} x^(3*k-1)/Product_{j=k..2*k-1} (1-x^j). - _Seiichi Manyama_, May 17 2023
%e a(8) = 3 counts these partitions: 53, 332, 11111111.
%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 *)
%o (PARI) my(N=70, x='x+O('x^N)); Vec(x+sum(k=1, N, x^(3*k-1)/prod(j=k, 2*k-1, 1-x^j))) \\ _Seiichi Manyama_, May 17 2023
%Y Cf. A118096, A237828.
%Y Cf. A237757, A237825, A237826, A237827, A000041.
%K nonn,easy
%O 1,5
%A _Clark Kimberling_, Feb 16 2014
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