|
|
A209318
|
|
Number T(n,k) of partitions of n with k parts in which no part occurs more than twice; triangle T(n,k), n>=0, 0<=k<=A055086(n), read by rows.
|
|
11
|
|
|
1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 5, 3, 0, 1, 4, 6, 4, 1, 0, 1, 5, 8, 6, 2, 0, 1, 5, 10, 8, 3, 0, 1, 6, 11, 12, 5, 1, 0, 1, 6, 14, 14, 8, 1, 0, 1, 7, 16, 19, 11, 3, 0, 1, 7, 18, 23, 16, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,12
|
|
LINKS
|
|
|
EXAMPLE
|
T(8,3) = 5: [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2].
T(8,4) = 3: [4,2,1,1], [3,3,1,1], [3,2,2,1].
T(9,3) = 6: [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2].
T(9,4) = 4: [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1].
T(9,5) = 1: [3,2,2,1,1].
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 2, 2;
0, 1, 3, 2, 1;
0, 1, 3, 4, 1;
0, 1, 4, 5, 3;
0, 1, 4, 6, 4, 1;
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(expand(b(n-i*j, i-1)*x^j), j=0..min(2, n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);
|
|
MATHEMATICA
|
max = 15; g = -1+Product[1+t*x^j+t^2*x^(2j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; t[0, 0] = 1; Table[Table[t[n, k], {k, 0, n}] /. {a__, 0 ..} -> {a}, {n, 0, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)
|
|
CROSSREFS
|
Columns k=0-10 give: A000007, A057427, A004526, A230059 (conjectured), A320592, A320593, A320594, A320595, A320596, A320597, A320598.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|