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A237831 Number of partitions of n such that (greatest part) - (least part) <= number of parts.

%I A237831 #27 Dec 18 2023 10:09:14
%S A237831 1,2,3,5,6,10,12,18,23,32,40,57,70,94,120,157,196,256,318,408,508,640,
%T A237831 792,996,1223,1518,1863,2296,2798,3432,4162,5070,6130,7422,8936,10777,
%U A237831 12916,15500,18522,22136,26348,31376,37222,44160,52236,61756,72824,85847
%N A237831 Number of partitions of n such that (greatest part) - (least part) <= number of parts.
%H A237831 Seiichi Manyama, <a href="/A237831/b237831.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..93 from R. J. Mathar)
%H A237831 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016.
%F A237831 a(n) + A237833(n) = A000041(n). - _R. J. Mathar_, Nov 24 2017
%F A237831 G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * k * ( x^(k*(3*k-1)/2) + x^(k*(3*k+1)/2) ). (See Andrews' preprint.) - _Seiichi Manyama_, May 20 2023
%e A237831 a(6) = 10 counts all the 11 partitions of 6 except 4+1+2.
%t A237831 z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
%t A237831 Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
%t A237831 Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
%t A237831 Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
%t A237831 Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
%t A237831 Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
%o A237831 (PARI) my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*k*(x^(k*(3*k-1)/2)+x^(k*(3*k+1)/2)))) \\ _Seiichi Manyama_, May 20 2023
%Y A237831 Cf. A109083, A237830, A237832, A237833, A237834.
%K A237831 nonn,easy
%O A237831 1,2
%A A237831 _Clark Kimberling_, Feb 16 2014