OFFSET
0,6
COMMENTS
The definition forbids partitions with a part equal to 1, so the smallest possible part is 2, which however can appear at most once.
Note that considering partitions in standard decreasing order, we obtain A064428.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Giovanni Resta)
FORMULA
a(n) = Sum_{k>=0} A238406(n,k). - Alois P. Heinz, Feb 26 2014
a(n) = A238352(n,0). - Alois P. Heinz, Jun 08 2014
EXAMPLE
a(6) = 3, because of the 11 partitions of 6 only 3 do not contain a 1 in position 1, a 2 in position 2, or a 3 in position 3, namely (3,3), (2,4) and (6).
From Joerg Arndt, Mar 23 2014: (Start)
There are a(15) = 22 such partitions of 15:
01: [ 2 3 4 6 ]
02: [ 2 3 5 5 ]
03: [ 2 3 10 ]
04: [ 2 4 4 5 ]
05: [ 2 4 9 ]
06: [ 2 5 8 ]
07: [ 2 6 7 ]
08: [ 2 13 ]
09: [ 3 3 4 5 ]
10: [ 3 3 9 ]
11: [ 3 4 8 ]
12: [ 3 5 7 ]
13: [ 3 6 6 ]
14: [ 3 12 ]
15: [ 4 4 7 ]
16: [ 4 5 6 ]
17: [ 4 11 ]
18: [ 5 5 5 ]
19: [ 5 10 ]
20: [ 6 9 ]
21: [ 7 8 ]
22: [ 15 ]
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
end:
a:= n-> (p-> add(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(a(n), n=0..70); # Alois P. Heinz, Feb 26 2014
MATHEMATICA
a[n_] := Length@ Select[ IntegerPartitions@n, 0 < Min@ Abs[ Reverse@# - Range@ Length@#] &]; Array[a, 30]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]]; a[n_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Exponent[p, x]} ] ][b[n, n]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, Feb 26 2014
STATUS
approved