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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.

%I A182485 #28 Oct 08 2018 19:32:06
%S A182485 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,
%T A182485 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,
%U A182485 0,2,62,89,0,6,65,100,1,0,4,68,131,5,0,4,87
%N 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.
%H A182485 Alois P. Heinz, <a href="/A182485/b182485.txt">Rows n = 0..120, flattened</a>
%e A182485 T(0,0) = 1: [].
%e A182485 T(1,1) = 1: [1].
%e A182485 T(2,1) = 2: [1,1], [2].
%e A182485 T(4,1) = 3: [1,1,1,1], [2,2], [4].
%e A182485 T(4,2) = 1: [2,1,1]; part 2 occurs once and part 1 occurs twice.
%e A182485 T(5,2) = 3: [2,1,1,1], [2,2,1], [3,1,1].
%e A182485 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].
%e A182485 T(10,1) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
%e A182485 T(10,3) = 1: [3,2,2,1,1,1].
%e A182485 Triangle T(n,k) begins:
%e A182485   1;
%e A182485   0, 1;
%e A182485   0, 2;
%e A182485   0, 2;
%e A182485   0, 3,  1;
%e A182485   0, 2,  3;
%e A182485   0, 4,  3;
%e A182485   0, 2,  8;
%e A182485   0, 4,  9;
%e A182485   0, 3, 12;
%e A182485   0, 4, 16, 1;
%p A182485 b:= proc(n, i, t, s) option remember;
%p A182485       `if`(nops(s)>t, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t, s)+
%p A182485       add(`if`(j in s, 0, b(n-i*j, i-1, t, s union {j})), j=1..n/i))))
%p A182485     end:
%p A182485 g:= proc(n) local i; for i while i*(i+1)*(i+2)/6<=n do od; i-1 end:
%p A182485 T:= n-> seq(b(n, n, k, {}) -b(n, n, k-1, {}), k=0..g(n)):
%p A182485 seq(T(n), n=0..30);
%t A182485 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 *)
%Y A182485 Columns k=0-7 give: A000007, A000005, A182473, A212114, A212115, A212116, A212117, A212118.
%Y A182485 Row sums give: A098859.
%Y A182485 First row with length (t+1): A000292(t).
%Y A182485 Cf. A242896 (the same for compositions):
%K A182485 nonn,tabf
%O A182485 0,5
%A A182485 _Alois P. Heinz_, May 01 2012