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A237976 Number of strict partitions of n such that (least part) < number of parts.

%I A237976 #24 Jan 18 2022 06:42:24
%S A237976 0,0,0,1,1,1,2,2,3,4,6,7,9,11,14,17,21,25,31,37,45,54,64,76,90,106,
%T A237976 124,146,170,198,230,267,308,357,410,472,542,621,709,811,923,1051,
%U A237976 1194,1355,1534,1738,1962,2215,2497,2812,3161,3553,3986,4469,5005,5600,6258
%N A237976 Number of strict partitions of n such that (least part) < number of parts.
%H A237976 Seiichi Manyama, <a href="/A237976/b237976.txt">Table of n, a(n) for n = 0..1000</a>
%F A237976 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k*(k-1))) / Product_{j=1..k} (1-x^j). - _Seiichi Manyama_, Jan 13 2022
%F A237976 a(n) = A000009(n) - A025157(n). - _Vaclav Kotesovec_, Jan 18 2022
%e A237976 a(8) = 3 counts these partitions:  71, 521, 431.
%t A237976 z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A237976 p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
%t A237976 Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
%t A237976 Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
%t A237976 Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
%t A237976 Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
%t A237976 Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)
%o A237976 (PARI) my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^(k*(k-1)))/prod(j=1, k, 1-x^j)))) \\ _Seiichi Manyama_, Jan 13 2022
%Y A237976 Cf. A000009, A237977, A096401, A237979, A025157, A039899.
%K A237976 nonn,easy
%O A237976 0,7
%A A237976 _Clark Kimberling_, Feb 18 2014
%E A237976 Prepended a(0)=0, _Seiichi Manyama_, Jan 13 2022