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A182485
Number of partitions of n into exactly k different parts with distinct multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.
9
1, 0, 1, 0, 2, 0, 2, 0, 3, 1, 0, 2, 3, 0, 4, 3, 0, 2, 8, 0, 4, 9, 0, 3, 12, 0, 4, 16, 1, 0, 2, 22, 4, 0, 6, 20, 5, 0, 2, 31, 12, 0, 4, 35, 16, 0, 4, 34, 24, 0, 5, 44, 33, 0, 2, 51, 52, 0, 6, 53, 57, 0, 2, 62, 89, 0, 6, 65, 100, 1, 0, 4, 68, 131, 5, 0, 4, 87
OFFSET
0,5
LINKS
EXAMPLE
T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,1], [2].
T(4,1) = 3: [1,1,1,1], [2,2], [4].
T(4,2) = 1: [2,1,1]; part 2 occurs once and part 1 occurs twice.
T(5,2) = 3: [2,1,1,1], [2,2,1], [3,1,1].
T(7,2) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1].
T(10,1) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
T(10,3) = 1: [3,2,2,1,1,1].
Triangle T(n,k) begins:
1;
0, 1;
0, 2;
0, 2;
0, 3, 1;
0, 2, 3;
0, 4, 3;
0, 2, 8;
0, 4, 9;
0, 3, 12;
0, 4, 16, 1;
MAPLE
b:= proc(n, i, t, s) option remember;
`if`(nops(s)>t, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t, s)+
add(`if`(j in s, 0, b(n-i*j, i-1, t, s union {j})), j=1..n/i))))
end:
g:= proc(n) local i; for i while i*(i+1)*(i+2)/6<=n do od; i-1 end:
T:= n-> seq(b(n, n, k, {}) -b(n, n, k-1, {}), k=0..g(n)):
seq(T(n), n=0..30);
MATHEMATICA
b[n_, i_, t_, s_] := b[n, i, t, s] = If[Length[s] > t, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, t, s] + Sum[If[MemberQ[s, j], 0, b[n-i*j, i-1, t, s ~Union~ {j}]], {j, 1, n/i}]]]]; g[n_] := Module[{i}, For[ i = 1, i*(i+1)*(i+2)/6 <= n , i++]; i-1 ]; t[n_] := Table [b[n, n, k, {}] - b[n, n, k-1, {}], {k, 0, g[n]}]; Table [t[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)
CROSSREFS
Row sums give: A098859.
First row with length (t+1): A000292(t).
Cf. A242896 (the same for compositions):
Sequence in context: A094098 A326844 A261079 * A164960 A124137 A164092
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, May 01 2012
STATUS
approved