cp's OEIS Frontend

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.

%I A209318 #28 Oct 16 2018 16:35:33
%S A209318 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,
%T A209318 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,
%U A209318 7,16,19,11,3,0,1,7,18,23,16,5
%N 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.
%H A209318 Alois P. Heinz, <a href="/A209318/b209318.txt">Rows n = 0..400, flattened</a>
%e A209318 T(8,3) = 5: [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2].
%e A209318 T(8,4) = 3: [4,2,1,1], [3,3,1,1], [3,2,2,1].
%e A209318 T(9,3) = 6: [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2].
%e A209318 T(9,4) = 4: [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1].
%e A209318 T(9,5) = 1: [3,2,2,1,1].
%e A209318 Triangle begins:
%e A209318   1;
%e A209318   0, 1;
%e A209318   0, 1, 1;
%e A209318   0, 1, 1;
%e A209318   0, 1, 2, 1;
%e A209318   0, 1, 2, 2;
%e A209318   0, 1, 3, 2, 1;
%e A209318   0, 1, 3, 4, 1;
%e A209318   0, 1, 4, 5, 3;
%e A209318   0, 1, 4, 6, 4, 1;
%p A209318 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A209318       add(expand(b(n-i*j, i-1)*x^j), j=0..min(2, n/i))))
%p A209318     end:
%p A209318 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A209318 seq(T(n), n=0..20);
%t A209318 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 *)
%Y A209318 Columns k=0-10 give: A000007, A057427, A004526, A230059 (conjectured), A320592, A320593, A320594, A320595, A320596, A320597, A320598.
%Y A209318 Row sums give: A000726.
%Y A209318 Row lengths give: A000267.
%Y A209318 Cf. A002620, A008289 (no part more than once), A055086, A117147 (no part more than 3 times).
%K A209318 nonn,tabf,look
%O A209318 0,12
%A A209318 _Alois P. Heinz_, Jan 19 2013