A212544 Number of partitions of n containing at least one part m-4 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 18, 24, 37, 48, 69, 89, 122, 155, 207, 259, 337, 419, 534, 657, 827, 1008, 1252, 1518, 1864, 2246, 2736, 3276, 3960, 4722, 5668, 6727, 8032, 9492, 11274, 13279, 15696, 18424, 21694, 25380, 29772, 34736, 40604, 47244, 55060, 63897

Offset

4,5

Links

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Formula

G.f.: Sum_{i>0} x^(2*i+4) / Product_{j=1..4+i} (1-x^j).

Example

a(6) = 1: [5,1].
a(7) = 1: [5,1,1].
a(8) = 3: [5,1,1,1], [5,2,1], [6,2].
a(9) = 4: [5,1,1,1,1], [5,2,1,1], [5,3,1], [6,2,1].
a(10) = 8: [5,1,1,1,1,1], [5,2,1,1,1], [5,2,2,1], [5,3,1,1], [5,4,1], [6,2,1,1], [6,2,2], [7,3].

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-4, min(n-2*m-4, m+4)), m=1..(n-4)/2):
seq(a(n), n=4..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 - 4, Min[n - 2 m - 4, m + 4]], {m, 1, (n - 4)/2}];
a /@ Range[4, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A212551.

Sequence in context: A099108 A208971 A001994 * A349801 A320787 A183151

Adjacent sequences: A212541 A212542 A212543 * A212545 A212546 A212547

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved