A058399 Triangle of partial row sums of partition triangle A008284.

1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1, 11, 10, 7, 4, 2, 1, 15, 14, 11, 7, 4, 2, 1, 22, 21, 17, 12, 7, 4, 2, 1, 30, 29, 25, 18, 12, 7, 4, 2, 1, 42, 41, 36, 28, 19, 12, 7, 4, 2, 1, 56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1, 77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1, 101, 100, 94, 80, 62

Offset: 1

Author

Wolfdieter Lang, Dec 11 2000

Keywords

• nonn
• tabl

Comments

T(n,m) is also the number of m-th largest elements in all partitions of n. - Omar E. Pol, Feb 14 2012

It appears that reversed rows converge to A000070. - Omar E. Pol, Mar 10 2012

The row sums give A006128. - Omar E. Pol, Mar 26 2012

T(n,m) is also the number of regions traversed by the m-th column of the section model of partitions with n sections (Cf. A135010, A206437). - Omar E. Pol, Apr 20 2012

Examples

			From _Omar E. Pol_, Mar 10 2012: (Start)
Triangle begins:
   1;
   2,  1;
   3,  2,  1;
   5,  4,  2,  1;
   7,  6,  4,  2,  1;
  11, 10,  7,  4,  2,  1;
  15, 14, 11,  7,  4,  2,  1;
  22, 21, 17, 12,  7,  4,  2,  1;
  30, 29, 25, 18, 12,  7,  4,  2,  1;
  42, 41, 36, 28, 19, 12,  7,  4,  2,  1;
  56, 55, 50, 40, 29, 19, 12,  7,  4,  2,  1;
  77, 76, 70, 58, 43, 30, 19, 12,  7,  4,  2,  1;
(End)
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

Columns 1-5: A000041(n), A000065(n+1), A004250(n+2), A035300(n-1), A035301(n-1), n >= 1.

Cf. A008284.

Programs

Maple

b:= proc(n, k) option remember;
      `if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
    end:
T:= (n, m)-> b(n,n) -b(n,m-1):
seq (seq (T(n, m), m=1..n), n=1..15);  # Alois P. Heinz, Apr 20 2012

Mathematica

t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)

Formula

T(n, m) = Sum_{k=m..n} A008284(n, k).

G.f. for m-th column: Sum_{n>=1} x^(n)/Product_{k=1..n+m-1} (1 - x^k).

T(n, m) = Sum_{k=1..n} A207379(k, m). - Omar E. Pol, Apr 22 2012