A212550 Number of partitions of n containing at least one part m-10 if m is the largest part.

0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 159, 211, 291, 381, 512, 663, 873, 1117, 1448, 1833, 2342, 2938, 3708, 4611, 5760, 7105, 8792, 10769, 13215, 16077, 19585, 23679, 28651, 34447, 41424, 49541, 59248, 70509, 83892, 99390, 117695, 138846, 163708

Offset: 10

Author

Alois P. Heinz, May 20 2012

Keywords

• nonn

Examples

			a(12) = 1: [11,1].
a(13) = 1: [11,1,1].
a(14) = 3: [11,1,1,1], [11,2,1], [12,2].
a(15) = 4: [11,1,1,1,1], [11,2,1,1], [11,3,1], [12,2,1].
a(16) = 8: [11,1,1,1,1,1], [11,2,1,1,1], [11,2,2,1], [11,3,1,1], [11,4,1], [12,2,1,1], [12,2,2], [13,3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 10..1000

Crossrefs

Column k=10 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-10, min(n-2*m-10, m+10)), m=1..(n-10)/2):
seq(a(n), n=10..60);

Mathematica

Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]-10]&)],{n,10,60}] (* _Harvey P. Dale, Feb 10 2015 *)
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 - 2m - 10, Min[n - 2m - 10, m + 10]], {m, 1, (n - 10)/2}];
a /@ Range[10, 60] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

G.f.: Sum_{i>0} x^(2*i+10) / Product_{j=1..10+i} (1-x^j).