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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
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.
Row sums give: A000726.
Row lengths give: A000267.
Cf. A002620, A008289 (no part more than once), A055086, A117147 (no part more than 3 times).
Sequence in context: A374212 A303903 A348515 * A170984 A114021 A239287
KEYWORD
nonn,tabf,look
AUTHOR
Alois P. Heinz, Jan 19 2013
STATUS
approved