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A237830 Number of partitions of n such that (greatest part) - (least part) < number of parts.

%I A237830 #27 Dec 18 2023 10:10:28
%S A237830 1,2,3,4,6,8,11,15,20,27,36,47,62,81,105,135,174,222,282,357,450,565,
%T A237830 707,880,1093,1353,1669,2052,2517,3077,3753,4565,5539,6704,8097,9755,
%U A237830 11730,14075,16854,20142,24029,28611,34009,40355,47807,56542,66772,78728
%N A237830 Number of partitions of n such that (greatest part) - (least part) < number of parts.
%H A237830 Seiichi Manyama, <a href="/A237830/b237830.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..95 from R. J. Mathar)
%H A237830 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 A237830 a(n) + A237834(n) = A000041(n). - _R. J. Mathar_, Nov 24 2017
%F A237830 G.f.: (1/Product_{k>=1} (1-x^k)) * (x/(1-x)) * Sum_{k>=1} (-1)^(k-1) * x^(3*k*(k-1)/2) * (1-x^(2*k)). (See Andrews' preprint.) - _Seiichi Manyama_, May 20 2023
%e A237830 a(6) = 8 counts these partitions:  6, 3+3, 4+1+1, 3+2+1, 2+2+2, 3+1+1+1, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1.
%t A237830 z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
%t A237830 Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
%t A237830 Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
%t A237830 Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
%t A237830 Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
%t A237830 Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
%o A237830 (PARI) my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*x/(1-x)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k-1)/2)*(1-x^(2*k)))) \\ _Seiichi Manyama_, May 20 2023
%Y A237830 Cf. A000070, A237831-A237834.
%K A237830 nonn,easy
%O A237830 1,2
%A A237830 _Clark Kimberling_, Feb 16 2014