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Number of partitions p of n into distinct parts such that max(p) > 1 + 2*min(p).
3

%I #10 Sep 25 2024 12:08:36

%S 0,0,0,0,0,1,1,2,4,4,6,8,11,13,16,21,26,32,38,45,56,66,79,94,110,128,

%T 151,178,207,240,277,320,370,426,488,561,642,732,834,948,1079,1225,

%U 1388,1570,1774,2002,2254,2540,2856,3206,3598,4034,4516,5050,5642,6298

%N Number of partitions p of n into distinct parts such that max(p) > 1 + 2*min(p).

%F a(n) + A241061(n) + A241062(n) = A000009(n) for n >= 1.

%F a(n) = A241037(n) - A241062(n) for n>= 0.

%e a(10) counts these 6 partitions: 91, 82, 721, 631, 541, 4321

%t z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];

%t Table[Count[f[n], p_ /; Max[p] < 1 + 2*Min[p]], {n, 0, z}] (* A241061 *)

%t Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Min[p]], {n, 0, z}] (* A207642 *)

%t Table[Count[f[n], p_ /; Max[p] == 1 + 2*Min[p]], {n, 0, z}] (* A241062 *)

%t Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Min[p]], {n, 0, z}] (* A241037 *)

%t Table[Count[f[n], p_ /; Max[p] > 1 + 2*Min[p]], {n, 0, z}] (* A241064 *)

%t Table[Count[IntegerPartitions[n],_?(Length[#]==Length[Union[#]]&&#[[1]]>2#[[-1]]+1&)],{n,0,60}]//Quiet (* _Harvey P. Dale_, Sep 25 2024 *)

%Y Cf. A207642, A241061, A241062, A241064, A000009.

%K nonn,easy

%O 0,8

%A _Clark Kimberling_, Apr 16 2014