A212547 Number of partitions of n containing at least one part m-7 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 55, 81, 108, 152, 199, 272, 351, 467, 596, 776, 979, 1255, 1566, 1978, 2448, 3054, 3747, 4628, 5635, 6896, 8342, 10125, 12172, 14673, 17537, 21005, 24981, 29748, 35210, 41718, 49161, 57974, 68049, 79902, 93440, 109295

Offset

7,5

Links

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Formula

G.f.: Sum_{i>0} x^(2*i+7) / Product_{j=1..7+i} (1-x^j).

Example

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

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-7, min(n-2*m-7, m+7)), m=1..(n-7)/2):
seq(a(n), n=7..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 - 7, Min[n - 2 m - 7, m + 7]], {m, 1, (n - 7)/2}];
a /@ Range[7, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Crossrefs

Column k=7 of A212551.

Sequence in context: A357878 A358910 A212546 * A212548 A212549 A212550

Adjacent sequences: A212544 A212545 A212546 * A212548 A212549 A212550

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved