A212551 Number of partitions T(n,k) of n containing at least one other part m-k if m is the largest part; triangle T(n,k), n>=0, 0<=k<=n.

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 3, 1, 1, 0, 0, 4, 3, 3, 1, 1, 0, 0, 4, 6, 4, 3, 1, 1, 0, 0, 7, 7, 7, 4, 3, 1, 1, 0, 0, 8, 11, 9, 8, 4, 3, 1, 1, 0, 0, 12, 13, 15, 10, 8, 4, 3, 1, 1, 0, 0, 14, 20, 18, 17, 11, 8, 4, 3, 1, 1, 0, 0

Offset: 0

Author

Alois P. Heinz, May 20 2012

Keywords

• nonn
• tabl

Comments

Reversed rows converge to A024786.

Examples

			T(4,0) = 2: [1,1,1,1], [2,2].
T(4,1) = 1: [2,1,1].
T(5,1) = 3: [2,1,1,1], [2,2,1], [3,2].
T(6,2) = 3: [3,1,1,1], [3,2,1], [4,2].
T(7,2) = 4: [3,1,1,1,1], [3,2,1,1], [3,3,1], [4,2,1].
T(8,4) = 3: [5,1,1,1], [5,2,1], [6,2].
Triangle T(n,k) begins:
1;
0, 0;
1, 0, 0;
1, 1, 0, 0;
2, 1, 1, 0, 0;
2, 3, 1, 1, 0, 0;
4, 3, 3, 1, 1, 0, 0;
4, 6, 4, 3, 1, 1, 0, 0;
7, 7, 7, 4, 3, 1, 1, 0, 0;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Kronecker delta

Crossrefs

Columns k=0-10 give: A002865, A083751(n+1), A119907, A212543, A212544, A212545, A212546, A212547, A212548, A212549, A212550.

Row sums give A000070(n-2) for n>1.

Cf. A024786.

Programs

Maple

b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
    end:
T:= (n, k)-> `if`(n=0 and k=0, 1,
    add(b(n-2*m-k, min(n-2*m-k, m+k)), m=1..(n-k)/2)):
seq(seq(T(n, k), k=0..n), n=0..14);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + If[i > n, 0, b[n-i, i]]]; t[n_, k_] := If[n == 0 && k == 0, 1, Sum[b[n-2*m-k, Min[n-2*m-k, m+k]], {m, 1, (n-k)/2}]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

Formula

G.f. of column k: delta_{0,k} + Sum_{i>0} x^(2*i+k) / Product_{j=1..k+i} (1-x^j), where delta is the Kronecker delta.