A240448 - OEIS

%I #6 Apr 06 2020 20:28:52

%S 0,0,1,1,2,3,5,7,9,13,20,28,34,46,64,89,107,144,183,247,295,391,491,

%T 647,747,974,1200,1552,1815,2320,2778,3541,4104,5180,6191,7775,8913,

%U 11129,13178,16351,18754,23141,27024,33233,38036,46535,54202,66012,74903

%N Number of partitions p of n such that (sum of parts with multiplicity 1) < (sum of all other parts).

%F a(n) + A240452(n) = A000041(n) for n >= 0.

%e a(6) counts these 5 partitions:  33, 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. A240449, A240447, A240451, A240452, A000041.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Apr 05 2014