A218906 Number of different kernels of integer partitions of n.

1, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 27, 32, 37, 42, 48, 55, 63, 71, 80, 91, 103, 115, 129, 145, 162, 180, 200, 223, 248, 274, 303, 336, 371, 408, 449, 495, 544, 596, 653, 716, 784, 856, 934, 1021, 1114, 1212, 1319, 1436, 1561, 1694, 1838, 1995

Offset

1,3

Comments

The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.

It is also a partition of an integer (called the size of the kernel), always self-conjugate.

In fact, this sequence is the cumulative sum of A000700, the number of self-conjugate partitions of n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Formula

G.f.: -1/(1 - x) + (1/(1 - x))*Product_{k>=1} (1 + x^(2*k-1)). - Ilya Gutkovskiy, Dec 25 2016

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i-2))))
    end:
a:= proc(n) a(n):= b(n, n-1+irem(n, 2))+`if`(n=1, 0, a(n-1)) end:
seq (a(n), n=1..100);  # Alois P. Heinz, Nov 09 2012

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n, 0, b[n-i, i-2]]]]; a[n_] := b[n, n-1 + Mod[n, 2]] + If[n==1, 0, a[n-1]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 12 2015, after Alois P. Heinz *)

Crossrefs

Cf. A218904.

Cf. A000700.

Sequence in context: A112402 A056864 A029032 * A059809 A327634 A121492

Adjacent sequences: A218903 A218904 A218905 * A218907 A218908 A218909

Keyword

nonn

Author

Olivier Gérard, Nov 08 2012

Status

approved