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

1,2

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

Links

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

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

Example

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)

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 *)

Crossrefs

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

Cf. A008284.

Sequence in context: A119441 A347227 A322083 * A209434 A207611 A320973

Adjacent sequences: A058396 A058397 A058398 * A058400 A058401 A058402

Keyword

nonn,tabl

Author

Wolfdieter Lang, Dec 11 2000

Status

approved