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A237977 Number of strict partitions of n such that (least part) <= number of parts.

%I A237977 #22 Jan 18 2022 06:49:42
%S A237977 0,1,0,1,1,2,3,3,4,5,7,8,11,13,17,20,25,29,36,42,51,60,72,84,100,117,
%T A237977 137,160,187,216,251,290,334,385,442,507,581,664,757,864,982,1116,
%U A237977 1266,1435,1622,1835,2069,2333,2626,2954,3316,3724,4172,4673,5227,5844
%N A237977 Number of strict partitions of n such that (least part) <= number of parts.
%H A237977 Seiichi Manyama, <a href="/A237977/b237977.txt">Table of n, a(n) for n = 0..1000</a>
%F A237977 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * (1-x^(k^2)) / Product_{j=1..k} (1-x^j). - _Seiichi Manyama_, Jan 13 2022
%F A237977 a(n) = A000009(n) - A237979(n). - _Vaclav Kotesovec_, Jan 18 2022
%e A237977 a(8) = 4 counts these partitions:  71, 53, 521, 431.
%t A237977 z = 50; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A237977 p1[p_] := p1[p] = DeleteDuplicates[p]; t[p_] := t[p] = Length[p1[p]]
%t A237977 Table[Count[q[n], p_ /; Min[p] < t[p]], {n, z}]  (* A237976 *)
%t A237977 Table[Count[q[n], p_ /; Min[p] <= t[p]], {n, z}] (* A237977 *)
%t A237977 Table[Count[q[n], p_ /; Min[p] == t[p]], {n, z}] (* A096401 *)
%t A237977 Table[Count[q[n], p_ /; Min[p] > t[p]], {n, z}]  (* A237979 *)
%t A237977 Table[Count[q[n], p_ /; Min[p] >= t[p]], {n, z}] (* A025157 *)
%o A237977 (PARI) my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^(k*(k+1)/2)*(1-x^k^2)/prod(j=1, k, 1-x^j)))) \\ _Seiichi Manyama_, Jan 13 2022
%Y A237977 Cf. A000009, A237976, A096401, A237979, A025157, A039900.
%K A237977 nonn,easy
%O A237977 0,6
%A A237977 _Clark Kimberling_, Feb 18 2014
%E A237977 Prepended a(0)=0, _Seiichi Manyama_, Jan 13 2022