%I #32 Sep 08 2021 08:59:34

%S 0,0,0,1,1,2,3,5,5,8,10,13,16,20,23,31,36,43,52,62,72,87,102,120,139,

%T 163,188,220,254,292,338,389,444,510,581,665,758,862,978,1111,1258,

%U 1422,1608,1814,2042,2302,2588,2908,3261,3655,4093,4580,5118,5714,6374

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

%C From _Omar E. Pol_, Mar 04 2017: (Start)

%C Partitions into distinct parts are sometimes called "strict partitions".

%C a(n) is also the number of partitions of n into distinct parts, which are not the partitions into (one or more) consecutive parts. (End)

%H Robert Israel, <a href="/A238007/b238007.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = A000009(n) - A001227(n). - _Omar E. Pol_, Mar 04 2017

%F a(n) = A238005(n)+A238006(n). - _R. J. Mathar_, Sep 08 2021

%e a(9) = 5 counts these partitions: 81, 72, 63, 621, 531.

%p spart:= proc(n, a,b,k) option remember;

%p # count strict partitions of n in exactly k parts with parts in [a,b]

%p if min(k,n) = 0 then if n=k then return 1 else return 0 fi fi;

%p if n < k*(2*a+k-1)/2 or n > k*(2*b-k+1)/2 then return 0 fi;

%p add (procname(n-x, a, x-1,k-1), x=a..min(n,b));

%p end proc:

%p f:= n -> add(add(add(spart(n-a-b,a+1,b-1,k-2),k=2..b-a),b=a+2..n),a=1..n-2):

%p map(f, [$1..100]); # _Robert Israel_, Mar 06 2017

%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, A238006.

%K nonn,easy

%O 1,6

%A _Clark Kimberling_, Feb 17 2014