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%I #15 Dec 25 2016 21:08:36
%S 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,
%T 115,129,145,162,180,200,223,248,274,303,336,371,408,449,495,544,596,
%U 653,716,784,856,934,1021,1114,1212,1319,1436,1561,1694,1838,1995
%N Number of different kernels of integer partitions of n.
%C The kernel of an integer partition is the intersection of its Ferrers diagram and of the Ferrers diagram of its conjugate.
%C It is also a partition of an integer (called the size of the kernel), always self-conjugate.
%C In fact, this sequence is the cumulative sum of A000700, the number of self-conjugate partitions of n.
%H Alois P. Heinz, <a href="/A218906/b218906.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: -1/(1 - x) + (1/(1 - x))*Product_{k>=1} (1 + x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 25 2016
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i-2))))
%p end:
%p a:= proc(n) a(n):= b(n, n-1+irem(n, 2))+`if`(n=1, 0, a(n-1)) end:
%p seq (a(n), n=1..100); # _Alois P. Heinz_, Nov 09 2012
%t 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_ *)
%Y Cf. A218904.
%Y Cf. A000700.
%K nonn
%O 1,3
%A _Olivier Gérard_, Nov 08 2012
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