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%I #6 Apr 06 2020 19:20:38
%S 1,0,1,1,3,3,6,7,12,13,22,28,42,46,69,89,125,144,202,247,334,391,525,
%T 647,852,974,1274,1552,1998,2320,2980,3541,4485,5180,6535,7775,9731,
%U 11129,13862,16351,20213,23141,28523,33233,40698,46535,56780,66012,80182
%N Number of partitions p of n such that (sum of parts with multiplicity 1) <= (sum of all other parts).
%F a(n) + A240451(n) = A000041(n) for n >= 0.
%e a(6) counts these 5 partitions: 33, 3111, 222, 2211, 21111, 111111.
%t z = 30; p[n_] := p[n] = IntegerPartitions[n]; f[p_] := f[p] = First[Transpose[p]];
%t ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] < n &], {n, 0, z}]] (* shows the partitions *)
%t Map[Length, t] (* A240448 *)
%t ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] <= n &], {n, 0, z}]] (* shows the partitions *)
%t Map[Length, t] (* A240449 *)
%t ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] == n &], {n, 0, z}]] (* shows the partitions *)
%t Map[Length, t] (* A240447 with alternating 0's *)
%t ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] > n &], {n, 0, z}]] (* shows the partitions *)
%t Map[Length, t] (* A240451 *)
%t ColumnForm[t = Table[Select[p[n], 2 Total[f[Select[#, Last[#] == 1 &] /. {} -> {{0, 0}}]] &[Tally[#]] >= n &], {n, 0, z}]] (* shows the partitions *)
%t Map[Length, t] (* A240452 *)
%t (* _Peter J. C. Moses_, Apr 02 2014 *)
%Y Cf. A240448, A240447, A240451, A240452, A000041.
%K nonn,easy
%O 0,5
%A _Clark Kimberling_, Apr 05 2014
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