A212548 Number of partitions of n containing at least one part m-8 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 82, 110, 156, 205, 281, 366, 488, 627, 821, 1041, 1340, 1684, 2135, 2657, 3331, 4108, 5095, 6238, 7663, 9315, 11354, 13709, 16588, 19915, 23936, 28580, 34154, 40573, 48225, 57031, 67452, 79428, 93530, 109695, 128639

Offset

8,5

Links

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Formula

G.f.: Sum_{i>0} x^(2*i+8) / Product_{j=1..8+i} (1-x^j).

Example

a(10) = 1: [9,1].
a(11) = 1: [9,1,1].
a(12) = 3: [9,1,1,1], [9,2,1], [10,2].
a(13) = 4: [9,1,1,1,1], [9,2,1,1], [9,3,1], [10,2,1].
a(14) = 8: [9,1,1,1,1,1], [9,2,1,1,1], [9,2,2,1], [9,3,1,1], [9,4,1], [10,2,1,1], [10,2,2], [11,3].

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-8, min(n-2*m-8, m+8)), m=1..(n-8)/2):
seq(a(n), n=8..60);

Mathematica

Table[Count[IntegerPartitions[n], _?(MemberQ[#, Max[#]-8]&)], {n, 8, 55}] (* Harvey P. Dale, May 05 2016 *)
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 - 8, Min[n - 2 m - 8, m + 8]], {m, 1, (n - 8)/2}];
a /@ Range[8, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Crossrefs

Column k=8 of A212551.

Sequence in context: A358910 A212546 A212547 * A212549 A212550 A024786

Adjacent sequences: A212545 A212546 A212547 * A212549 A212550 A212551

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved