A240180 - OEIS

%I #8 Apr 28 2014 15:02:35

%S 0,1,0,1,3,3,5,7,12,16,24,30,45,57,81,104,143,179,243,304,399,504,650,

%T 809,1039,1286,1622,2006,2508,3077,3822,4666,5747,6995,8552,10353,

%U 12603,15189,18371,22071,26570,31785,38104,45419,54213,64426,76596,90710

%N Number of partitions of n such that (least part) = (multiplicity of greatest part).

%C Also the number of partitions p of n such that min(p) = min(conjugate(p)). Example:a(7) counts these 7 partitions:  61, 511, 421, 4111, 3211, 31111, 211111, of which the respective conjugates are 211111, 31111, 3211, 4111, 421, 511, 61. - _Clark Kimberling_, Apr 11 2014

%F a(n) = A240179(n) - A240178(n), for n >= 0.

%F a(n) + 2*A240178(n) = A000041(n) for n >= 0.

%e a(6) counts these 5 partitions:  51, 411, 321, 3111, 21111.

%t z = 60; f[n_] := f[n] = IntegerPartitions[n];

%t Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}]  (* A240178 *)

%t Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)

%t Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)

%t Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240178, n>0 *)

%t Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)

%Y Cf. A240178, A240179, A000041.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Apr 02 2014