A212546 Number of partitions of n containing at least one part m-6 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 40, 54, 79, 104, 146, 190, 257, 330, 436, 552, 715, 896, 1140, 1415, 1777, 2184, 2711, 3308, 4063, 4922, 5995, 7214, 8720, 10435, 12525, 14910, 17793, 21076, 25016, 29507, 34850, 40941, 48148, 56351, 66007, 76995, 89855, 104484

Offset

6,5

Links

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Formula

G.f.: Sum_{i>0} x^(2*i+6) / Product_{j=1..6+i} (1-x^j).

Example

a(8) = 1: [7,1].
a(9) = 1: [7,1,1].
a(10) = 3: [7,1,1,1], [7,2,1], [8,2].
a(11) = 4: [7,1,1,1,1], [7,2,1,1], [7,3,1], [8,2,1].
a(12) = 8: [7,1,1,1,1,1], [7,2,1,1,1], [7,2,2,1], [7,3,1,1], [7,4,1], [8,2,1,1], [8,2,2], [9,3].

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-6, min(n-2*m-6, m+6)), m=1..(n-6)/2):
seq(a(n), n=6..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 - 6, Min[n - 2 m - 6, m + 6]], {m, 1, (n - 6)/2}];
a /@ Range[6, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Crossrefs

Column k=6 of A212551.

Sequence in context: A212545 A357878 A358910 * A212547 A212548 A212549

Adjacent sequences: A212543 A212544 A212545 * A212547 A212548 A212549

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved