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

3,5

Links

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

Formula

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

Example

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].

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 *)

Crossrefs

Column k=3 of A212551.

Sequence in context: A294085 A115264 A210631 * A355193 A238546 A147617

Adjacent sequences: A212540 A212541 A212542 * A212544 A212545 A212546

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved