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

10,5

Links

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

Formula

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

Example

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].

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 *)

Crossrefs

Column k=10 of A212551.

Sequence in context: A212547 A212548 A212549 * A024786 A299069 A097497

Adjacent sequences: A212547 A212548 A212549 * A212551 A212552 A212553

Keyword

nonn

Author

Alois P. Heinz, May 20 2012

Status

approved