A213177 Number T(n,k) of parts in all partitions of n with largest multiplicity k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

0, 0, 1, 0, 1, 2, 0, 3, 0, 3, 0, 3, 5, 0, 4, 0, 5, 6, 4, 0, 5, 0, 8, 9, 7, 5, 0, 6, 0, 10, 13, 13, 5, 6, 0, 7, 0, 13, 23, 14, 15, 6, 7, 0, 8, 0, 18, 30, 27, 16, 13, 7, 8, 0, 9, 0, 25, 44, 33, 30, 18, 15, 8, 9, 0, 10, 0, 30, 58, 55, 36, 34, 15, 17, 9, 10, 0, 11

Offset: 0

Author

Alois P. Heinz, Feb 27 2013

Keywords

• nonn
• tabl

Examples

			T(6,1) = 8: partitions of 6 with largest multiplicity 1 are [3,2,1], [4,2], [5,1], [6], with 3+2+2+1 = 8 parts.
T(6,2) = 9: [2,2,1,1], [3,3], [4,1,1].
T(6,3) = 7: [2,2,2], [3,1,1,1].
T(6,4) = 5: [2,1,1,1,1].
T(6,5) = 0.
T(6,6) = 6: [1,1,1,1,1,1].
Triangle begins:
  0;
  0,  1;
  0,  1,  2;
  0,  3,  0,  3;
  0,  3,  5,  0,  4;
  0,  5,  6,  4,  0,  5;
  0,  8,  9,  7,  5,  0,  6;
  0, 10, 13, 13,  5,  6,  0,  7;
  0, 13, 23, 14, 15,  6,  7,  0,  8;
  ...
		

Links

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

Crossrefs

Columns k=0-10 give: A000004, A015723, A320372, A320373, A320374, A320375, A320376, A320377, A320378, A320379, A320380.

Row sums give: A006128.

Main diagonal and first lower diagonal give: A001477, A063524.

T(2n,n) gives A320381.

Cf. A091602, A210485.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2] -b(n, n, k-1)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[, 0] = 0; T[n, k_] := b[n, n, k][[2]] - b[n, n, k-1][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Formula

T(n,k) = A210485(n,k) - A210485(n,k-1) for k>0, T(n,0) = 0.