A182701 Triangle T(n,k) = n*A000041(n-k) read by rows, 1 <= k <= n. Sum of the parts of all partitions of n that contain k as a part.

1, 2, 2, 6, 3, 3, 12, 8, 4, 4, 25, 15, 10, 5, 5, 42, 30, 18, 12, 6, 6, 77, 49, 35, 21, 14, 7, 7, 120, 88, 56, 40, 24, 16, 8, 8, 198, 135, 99, 63, 45, 27, 18, 9, 9, 300, 220, 150, 110, 70, 50, 30, 20, 10, 10, 462, 330, 242, 165, 121, 77, 55, 33, 22, 11, 11, 672, 504, 360, 264, 180, 132, 84, 60, 36, 24, 12, 12

Offset: 1

Author

Omar E. Pol, Nov 27 2010

Keywords

• nonn
• tabl

Comments

By definition, the entries in row n are divisible by n.

Row sums are 1, 4, 12, 28, 60, 114, ... = n*A000070(n).

Column 1 is A228816. - Omar E. Pol, Sep 25 2013

Examples

			Triangle begins:
    1;
    2,   2;
    6,   3,   3;
   12,   8,   4,   4;
   25,  15,  10,   5,   5;
   42,  30,  18,  12,   6,   6;
   77,  49,  35,  21,  14,   7,   7;
  120,  88,  56,  40,  24,  16,   8,   8;
  198, 135,  99,  63,  45,  27,  18,   9,   9;
  300, 220, 150, 110,  70,  50,  30,  20,  10,  10;
		

Links

• Robert Price, Table of n, a(n) for n = 1..5050 (First 100 rows)

Crossrefs

Cf. A000041, A027293, A066186, A135010, A138121, A182700, A182702.

Programs

Maple

A182701 := proc(n,k) n*combinat[numbpart](n-k) ; end proc:
seq(seq(A182701(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Nov 28 2010

Mathematica

T[n_, k_] := n PartitionsP[n - k];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 19 2019 *)

Formula

T(n,k) = A182700(n,k), 1 <= k < n.

T(n,k) = n*A027293(n,k). - Omar E. Pol, Sep 25 2013