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A237832 Number of partitions of n such that (greatest part) - (least part) = number of parts.

%I A237832 #27 Dec 18 2023 10:09:29
%S A237832 0,0,0,1,0,2,1,3,3,5,4,10,8,13,15,22,22,34,36,51,58,75,85,116,130,165,
%T A237832 194,244,281,355,409,505,591,718,839,1022,1186,1425,1668,1994,2319,
%U A237832 2765,3213,3805,4429,5214,6052,7119,8243,9645,11169,13026,15046,17511
%N A237832 Number of partitions of n such that (greatest part) - (least part) = number of parts.
%H A237832 Seiichi Manyama, <a href="/A237832/b237832.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..96 from R. J. Mathar)
%H A237832 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 A237832 A237830(n) + a(n) + A237833(n) = A000041(n). - _R. J. Mathar_, Nov 24 2017
%F A237832 G.f.: (1/Product_{k>=1} (1-x^k)) * (1/(1-x)) * Sum_{k>=1} (-1)^(k-1) * ( k * (1-x) * x^(k*(3*k-1)/2) * (1+x^k) - x^(3*k*(k-1)/2+1) * (1-x^(2*k)) ) - _Seiichi Manyama_, May 20 2023
%e A237832 a(6) = 2 counts these partitions:  4+2, 4+1+1.
%t A237832 z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
%t A237832 Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
%t A237832 Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
%t A237832 Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
%t A237832 Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
%t A237832 Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
%o A237832 (PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*1/(1-x)*sum(k=1, N, (-1)^(k-1)*(k*(1-x)*x^(k*(3*k-1)/2)*(1+x^k)-x^(3*k*(k-1)/2+1)*(1-x^(2*k)))))) \\ _Seiichi Manyama_, May 20 2023
%Y A237832 Cf. A237830, A237831, A237833, A237834.
%K A237832 nonn,easy
%O A237832 1,6
%A A237832 _Clark Kimberling_, Feb 16 2014