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

0,6

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) + 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

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

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

Sequence in context: A017842 A157725 A372592 * A378521 A372595 A182503

Adjacent sequences: A238391 A238392 A238393 * A238395 A238396 A238397

Keyword

nonn

Author

Giovanni Resta, Feb 26 2014

Status

approved