%I #12 Jun 05 2015 08:28:37

%S 1,0,0,3,4,6,8,8,9,22,22,34,50,60,74,105,120,144,186,234,280,358,440,

%T 524,665,782,954,1150,1354,1630,1944,2258,2666,3170,3728,4365,5128,

%U 5976,6978,8144,9488,10952,12700,14716,16932,19558,22434,25764,29505,33782

%N Number of partitions p of n such that exactly one number is in both p and its conjugate.

%C Second column of the array at A240181. Multiplicities greater than 1 are not counted; e.g. there is exactly one number that is in both {4,1,1} and {3,1,1,1}.

%H Manfred Scheucher, <a href="/A240675/b240675.txt">Table of n, a(n) for n = 1..65</a>

%H Manfred Scheucher, <a href="/A240675/a240675.sage.txt">Sage Script</a>

%e a(6) counts these 6 partitions: 51, 42, 411, 3111, 2211, 21111, of which the respective conjugates are 21111, 2211, 3111, 411, 42, 51.

%t z = 30; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; c[p_] := c[p] = Table[Count[#, _?(# >= i &)], {i, First[#]}] &[p]; b[n_] := b[n] = Table[Intersection[p[n, k], c[p[n, k]]], {k, 1, PartitionsP[n]}]; Table[Count[Map[Length, b[n]], 0], {n, 1, z}] (* A240674 *)

%t Table[Count[Map[Length, b[n]], 1], {n, 1, z}] (* A240675 *)

%o (PARI) conjug(v) = {my(m = matrix(#v, vecmax(v))); for (i=1, #v, for (j=1, v[i], m[i, j] = 1;);); vector(vecmax(v), i, sum(j=1, #v, m[j, i]));}

%o a(n) = {my(v = partitions(n)); my(nb = 0); for (k=1, #v, if (#setintersect(Set(v[k]), Set(conjug(v[k]))) == 1, nb++);); nb;} \\ _Michel Marcus_, Jun 02 2015

%Y Cf. A240674, A240181.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Apr 12 2014

%E More terms from _Manfred Scheucher_, Jun 01 2015