A237834 - OEIS

%I #19 Dec 18 2023 10:10:42

%S 0,0,0,1,1,3,4,7,10,15,20,30,39,54,71,96,123,163,208,270,342,437,548,

%T 695,865,1083,1341,1666,2048,2527,3089,3784,4604,5606,6786,8222,9907,

%U 11940,14331,17196,20554,24563,29252,34820,41327,49016,57982,68545,80833

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

%H R. J. Mathar, <a href="/A237834/b237834.txt">Table of n, a(n) for n = 1..95</a>

%H 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(n)+a(n) = A000041(n). - _R. J. Mathar_, Nov 24 2017

%e a(7) = 4 counts these partitions:  6+1, 5+2, 5+1+1, 4+2+1.

%t z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];

%t Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)

%t Table[Count[IntegerPartitions[n],_?(#[[1]]-#[[-1]]>=Length[#]&)],{n,50}] (* _Harvey P. Dale_, Jul 21 2023 *)

%Y Cf. A237830, A237831, A237832, A237833.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Feb 16 2014