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%I #16 Mar 16 2024 13:21:35
%S 0,1,1,1,1,2,1,2,2,2,2,3,3,3,3,4,4,5,4,5,6,6,7,7,7,8,9,10,10,11,11,12,
%T 14,14,15,17,17,18,19,20,23,24,25,26,28,29,31,34,35,37,40,42,44,46,48,
%U 51,55,58,61,64,67,70,75,77,82,87,90,96,101,105,111
%N Number of partitions p of n into distinct parts such that max(p) < 2*min(p).
%H John Tyler Rascoe, <a href="/A240874/b240874.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: Sum_{i>0} Sum_{j>=i} q^((i/2)*(i+(2*j)-1)) * q_binomial(i-1,j-i). - _John Tyler Rascoe_, Mar 16 2024
%e a(12) counts these 3 partitions: {12}, {7,5}, {5,4,3}.
%t z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t Table[Count[f[n], p_ /; Max[p] < 2*Min[p]], {n, 0, z}] (* this sequence *)
%t Table[Count[f[n], p_ /; Max[p] == 2*Min[p]], {n, 0, z}] (* A241035 *)
%t Table[Count[f[n], p_ /; Max[p] >= 2*Min[p]], {n, 0, z}] (* A241036 *)
%t Table[Count[f[n], p_ /; Max[p] > 2*Min[p]], {n, 0, z}] (* A241037 *)
%o (PARI)
%o p_q(k) = {prod(j=1,k, 1-q^j);}
%o GB_q(N,M)= {p_q(N+M)/(p_q(M)*p_q(N));}
%o A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=i, N-(i*(i+1)/2), q^((i/2)*(i+(2*j)-1)) * GB_q(i-1,j-i))));
%o concat([0],Vec(g))}
%o A_q(71) \\ _John Tyler Rascoe_, Mar 16 2024
%Y Cf. A241035, A241036, A241037.
%K nonn,easy
%O 0,6
%A _Clark Kimberling_, Apr 15 2014
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