A240449 - OEIS

%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