A238394 Number of partitions of n that sorted in increasing order do not contain a part k in position k.

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 8, 9, 12, 13, 17, 22, 28, 34, 42, 48, 59, 71, 88, 106, 130, 151, 181, 210, 250, 295, 354, 417, 494, 577, 675, 780, 909, 1053, 1231, 1431, 1668, 1930, 2240, 2573, 2963, 3392, 3896, 4461, 5129, 5873, 6742, 7710, 8816, 10043, 11439

Offset: 0

Author

Giovanni Resta, Feb 26 2014

Keywords

• nonn

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.

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Giovanni Resta)

Crossrefs

Cf. A000041, A238395, A064428, A001522, A238352.

Programs

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 *)

Formula

a(n) + A238395(n) = p(n) = A000041(n).

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