A212543 Number of partitions of n containing at least one part m-3 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 10, 17, 22, 33, 42, 60, 75, 103, 128, 169, 209, 271, 331, 421, 513, 642, 777, 963, 1158, 1421, 1703, 2070, 2471, 2985, 3546, 4257, 5043, 6019, 7105, 8443, 9933, 11752, 13790, 16247, 19012, 22326, 26052, 30492, 35500, 41420, 48108, 55980

Offset: 3

Author

Alois P. Heinz, May 20 2012

Keywords

• nonn

Examples

			a(5) = 1: [4,1].
a(6) = 1: [4,1,1].
a(7) = 3: [4,1,1,1], [4,2,1], [5,2].
a(8) = 4: [4,1,1,1,1], [4,2,1,1], [4,3,1], [5,2,1].
a(9) = 8: [4,1,1,1,1,1], [4,2,1,1,1], [4,2,2,1], [4,3,1,1], [4,4,1], [5,2,1,1], [5,2,2], [6,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..1000

Crossrefs

Column k=3 of A212551.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> add(b(n-2*m-3, min(n-2*m-3, m+3)), m=1..(n-3)/2):
seq(a(n), n=3..60);

Mathematica

Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]-3]&)],{n,3,60}] (* _Harvey P. Dale, Mar 01 2015 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i]]];
a[n_] := Sum[b[n - 2 m - 3, Min[n - 2 m - 3, m + 3]], {m, 1, (n - 3)/2}];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+3) / Product_{j=1..3+i} (1-x^j).