A238945 Number of partitions of n that have even-sized Ferrers matrix.

1, 0, 2, 2, 5, 5, 8, 9, 16, 19, 30, 37, 54, 66, 91, 112, 152, 187, 248, 307, 401, 495, 635, 781, 990, 1210, 1517, 1849, 2296, 2788, 3434, 4155, 5087, 6132, 7460, 8963, 10844, 12980, 15624, 18638, 22332, 26553

Offset

1,3

Comments

Also, the number of even numbers in row n of the array at A238943. Suppose that p is a partition of n, and let m = max{greatest part of p, number of parts of p}. Write the Ferrers graph of p with 1's as nodes, and pad the graph with 0's to form an m X m square matrix, which is introduced at A237981 as the Ferrers matrix of p, denoted by f(p). The size of f(p) is m.

Links

Table of n, a(n) for n=1..42.

Formula

a(n) + A238944(n) = A000041(n).

Example

(See the example at A238943.)

Mathematica

p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; a[t_] := Max[Max[t], Length[t]]; z = 42; t = Mod[Table[a[p[n, k]], {n, 1, z}, {k, 1, PartitionsP[n]}], 2];
u = Table[Count[t[[n]], 0], {n, 1, z}]  (* A238944 *)
v = Table[Count[t[[n]], 1], {n, 1, z}]  (* A238945 *)

Crossrefs

Cf. A237981, A238944, A238943, A000041.

Sequence in context: A035624 A340223 A073707 * A340572 A091609 A183563

Adjacent sequences: A238942 A238943 A238944 * A238946 A238947 A238948

Keyword

nonn,easy

Author

Clark Kimberling, Mar 07 2014

Status

approved