|
|
A212549
|
|
Number of partitions of n containing at least one part m-9 if m is the largest part.
|
|
2
|
|
|
0, 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 111, 158, 209, 287, 375, 503, 648, 852, 1086, 1403, 1770, 2255, 2817, 3546, 4393, 5469, 6723, 8294, 10120, 12382, 15011, 18228, 21965, 26497, 31749, 38069, 45383, 54114, 64204, 76176, 89975, 106259, 124998, 146987
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
9,5
|
|
LINKS
|
|
|
FORMULA
|
G.f.: Sum_{i>0} x^(2*i+9) / Product_{j=1..9+i} (1-x^j).
|
|
EXAMPLE
|
a(11) = 1: [10,1].
a(12) = 1: [10,1,1].
a(13) = 3: [10,1,1,1], [10,2,1], [11,2].
a(14) = 4: [10,1,1,1,1], [10,2,1,1], [10,3,1], [11,2,1].
a(15) = 8: [10,1,1,1,1,1], [10,2,1,1,1], [10,2,2,1], [10,3,1,1], [10,4,1], [11,2,1,1], [11,2,2], [12,3].
|
|
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-9, min(n-2*m-9, m+9)), m=1..(n-9)/2):
seq(a(n), n=9..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 - 9, Min[n - 2 m - 9, m + 9]], {m, 1, (n - 9)/2}];
Table[Count[IntegerPartitions[n], _?(MemberQ[#, #[[1]]-9]&)], {n, 9, 60}] (* Harvey P. Dale, Jun 08 2022 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|