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A218906 Number of different kernels of integer partitions of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 21 12:20 EST 2019. Contains 329370 sequences. (Running on oeis4.)