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
KEYWORD
nonn
AUTHOR
Olivier Gérard, Nov 08 2012
STATUS
approved