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

Author

Alois P. Heinz, May 20 2012

Keywords

• nonn

Examples

			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].
		

Links

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

Crossrefs

Column k=4 of A212551.

Programs

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 *)

Formula

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