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Number of partitions of n such that (greatest part) + (least part) > number of parts.
5

%I #5 Feb 22 2014 18:50:00

%S 1,2,2,4,5,8,10,16,20,28,37,51,65,88,112,147,187,243,305,391,488,618,

%T 769,963,1189,1479,1817,2241,2739,3357,4081,4976,6021,7296,8794,10605,

%U 12728,15284,18272,21845,26024,30996,36797,43671,51676,61118,72106,85013

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

%e a(6) = 8 counts these partitions: 6, 51, 42, 411, 33, 321, 222, 3111..

%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}] (* A237822 *)

%t Table[Count[q[n], p_ /; Max[p] + Min[p] <= t[p]], {n, z}] (* A237823 *)

%t Table[Count[q[n], p_ /; Max[p] + Min[p] == t[p]], {n, z}] (* A237869 *)

%t Table[Count[q[n], p_ /; Max[p] + Min[p] > t[p]], {n, z}] (* A237870 *)

%t Table[Count[q[n], p_ /; Max[p] + Min[p] >= t[p]], {n, z}] (* A237871 *)

%Y Cf. A237822, A237823, A237869, A237870.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 18 2014