A212549 Number of partitions of n containing at least one part m-9 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 111, 158, 209, 287, 375, 503, 648, 852, 1086, 1403, 1770, 2255, 2817, 3546, 4393, 5469, 6723, 8294, 10120, 12382, 15011, 18228, 21965, 26497, 31749, 38069, 45383, 54114, 64204, 76176, 89975, 106259, 124998, 146987

Offset

9,5

Links

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Formula

G.f.: Sum_{i>0} x^(2*i+9) / Product_{j=1..9+i} (1-x^j).

Example

a(11) = 1: [10,1].
a(12) = 1: [10,1,1].
a(13) = 3: [10,1,1,1], [10,2,1], [11,2].
a(14) = 4: [10,1,1,1,1], [10,2,1,1], [10,3,1], [11,2,1].
a(15) = 8: [10,1,1,1,1,1], [10,2,1,1,1], [10,2,2,1], [10,3,1,1], [10,4,1], [11,2,1,1], [11,2,2], [12,3].

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-9, min(n-2*m-9, m+9)), m=1..(n-9)/2):
seq(a(n), n=9..60);

Mathematica

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 - 9, Min[n - 2 m - 9, m + 9]], {m, 1, (n - 9)/2}];
a /@ Range[9, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n], _?(MemberQ[#, #[[1]]-9]&)], {n, 9, 60}] (* Harvey P. Dale, Jun 08 2022 *)

Crossrefs

Column k=9 of A212551.

Sequence in context: A212546 A212547 A212548 * A212550 A024786 A299069

Adjacent sequences: A212546 A212547 A212548 * A212550 A212551 A212552

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved