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Number of strict partitions of n such that (greatest part) - (least part) > (number of parts).
4

%I #18 Sep 12 2021 14:12:47

%S 0,0,0,0,1,1,2,3,5,6,8,11,14,18,22,27,33,41,49,59,70,83,98,116,136,

%T 159,186,215,249,289,333,383,441,505,578,660,752,856,974,1105,1252,

%U 1418,1602,1808,2039,2295,2581,2901,3255,3649,4088,4573,5111,5709,6368

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

%F A001227(n) + A238005(n) + a(n) = A000009(n). - _R. J. Mathar_, Sep 08 2021

%F From _Omar E. Pol_, Sep 11 2021: (Start)

%F a(n) = A000009(n) - A003056(n).

%F a(n) = A238007(n) - A238005(n). (End)

%e a(8) = 3 counts these partitions: 7+1, 6+2, 5+2+1.

%t z = 70; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]];

%t Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}] (* A001227 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A003056 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A238005 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}] (* A238006 *)

%t Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A238007 *)

%Y Cf. A000009, A001227, A003056, A238005, A238007.

%K nonn,easy

%O 1,7

%A _Clark Kimberling_, Feb 17 2014