A145574 Array a(n,m) for number of partitions of n>=2 with m parts having no part 1. Hence m=1..floor(n/2).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 3, 1, 1, 4, 4, 2, 1, 1, 4, 5, 3, 1, 1, 5, 7, 5, 2, 1, 1, 5, 8, 6, 3, 1, 1, 6, 10, 9, 5, 2, 1, 1, 6, 12, 11, 7, 3, 1, 1, 7, 14, 15, 10, 5, 2, 1, 1, 7, 16, 18, 13, 7, 3, 1, 1, 8, 19, 23, 18, 11, 5, 2, 1, 1, 8, 21, 27, 23, 14, 7, 3, 1, 1, 9, 24, 34, 30

Offset

2,8

Comments

The row lengths sequence is floor(n/2) = [1,1,2,2,3,3,4,4,...], see A008619(n-1), n>=2.

Obtained from the characteristic partition array A145573 by summing in row n>=2 over entries belonging to like parts number m.

The column sequences give A000012, A004526, A001399, A001400, A001401, A001402, A026813 for m=1..7.

Links

Alois P. Heinz, Rows n = 2..200, flattened

W. Lang and M. Sjodahl, First 20 rows of the array and row sums.

Formula

a(n,m) = sum over entries of A145573(n,k) array which belong to partitions with part number m, for m=1..floor(n/2)). Note that partitions with parts number m>floor(n/2) have always at least one part 1.

G.f.: Product_{i>=2} 1/(1- y*x^i). - Geoffrey Critzer, Sep 23 2012

Example

1;
1;
1, 1;
1, 1;
1, 2, 1;
1, 2, 1;
1, 3, 2, 1;
1, 3, 3, 1;
1, 4, 4, 2, 1;

Maple

b:= proc(n, i, t) option remember; `if`(2*t>n or t*i<n, 0,
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1, t) +b(n-i, i, t-1))))
    end:
a:= (n, m)-> b(n, n, m):
seq(seq(a(n, m), m=1..iquo(n, 2)), n=2..30); # Alois P. Heinz, Oct 18 2012

Mathematica

nn=15; f[list_]:=Select[list, #>0&]; p=Product[1/(1-y x^i), {i, 2, nn}]; Drop[Map[f, CoefficientList[Series[p, {x, 0, nn}], {x, y}]], 1]//Grid  (* Geoffrey Critzer, Sep 23 2012 *)

Prog

(Sage)  # Prints the table; cf. A011973.
for n in (2..20): [Partitions(n, length=m, min_part=2).cardinality() for m in (1..n//2)]  # Peter Luschny, Oct 18 2012

Crossrefs

Cf. A145573, A002865 (row sums).

Sequence in context: A108316 A322426 A335438 * A182579 A290737 A056138

Adjacent sequences: A145571 A145572 A145573 * A145575 A145576 A145577

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang and Malin Sjodahl, Mar 06 2009

Status

approved