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

Author

Olivier Gérard, Nov 08 2012

Keywords

• nonn

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

Crossrefs

Cf. A218904.

Cf. A000700.

Programs

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 *)

Formula

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