A181187 Triangle read by rows: T(n,k) = sum of k-th largest elements in all partitions of n.

1, 3, 1, 6, 2, 1, 12, 5, 2, 1, 20, 8, 4, 2, 1, 35, 16, 8, 4, 2, 1, 54, 24, 13, 7, 4, 2, 1, 86, 41, 22, 13, 7, 4, 2, 1, 128, 61, 35, 20, 12, 7, 4, 2, 1, 192, 95, 54, 33, 20, 12, 7, 4, 2, 1, 275, 136, 80, 49, 31, 19, 12, 7, 4, 2, 1, 399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1, 556, 284

Offset: 1

Author

Wouter Meeussen, Oct 09 2010

Keywords

• easy
• nonn
• tabl

Comments

For the connection with A066897 and A066898 see A206563. - Omar E. Pol, Feb 13 2012

T(n,k) is also the total number of parts >= k in all partitions of n. - Omar E. Pol, Feb 14 2012

The first differences of row n together with 1 give the row n of triangle A066633. - Omar E. Pol, Feb 26 2012

We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1. It appears that the sum of the k-th ranks of all partitions of n is equal to zero. - Omar E. Pol, Mar 04 2012

T(n,k) is also the total number of divisors >= k of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021

Examples

			From _Omar E. Pol_, Feb 13 2012: (Start)
Illustration of initial terms. First five rows of triangle as sums of columns from the partitions of the first five positive integers:
.
.                            5
.                            3+2
.                  4         4+1
.                  2+2       2+2+1
.          3       3+1       3+1+1
.     2    2+1     2+1+1     2+1+1+1
.  1  1+1  1+1+1   1+1+1+1   1+1+1+1+1
. -------------------------------------
.  1, 3,1, 6,2,1, 12,5,2,1, 20,8,4,2,1 --> This triangle
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|
.  1, 2,1, 4,1,1,  7,3,1,1, 12,4,2,1,1 --> A066633
.
For more information see A207031 and A206563.
...
Triangle begins:
    1;
    3,   1;
    6,   2,   1;
   12,   5,   2,  1;
   20,   8,   4,  2,  1;
   35,  16,   8,  4,  2,  1;
   54,  24,  13,  7,  4,  2,  1;
   86,  41,  22, 13,  7,  4,  2,  1;
  128,  61,  35, 20, 12,  7,  4,  2, 1;
  192,  95,  54, 33, 20, 12,  7,  4, 2, 1;
  275, 136,  80, 49, 31, 19, 12,  7, 4, 2, 1;
  399, 204, 121, 76, 48, 31, 19, 12, 7, 4, 2, 1;
(End)
		

Links

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

Crossrefs

Row sums are A066186. First column is A006128. Reverse of each row converges to A000070.

Columns 2-3: A096541, A207033. - Omar E. Pol, Feb 18 2012

T(2n,n) gives A216053(n+1).

Cf. A206283.

Programs

Maple

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0], b(n-i, i));
         p(p(f, g), [0$i, g[1]])
      fi
    end:
T:= proc(n) local j, l, r, t;
      l, r, t:= b(n, n), 1, 1;
      for j from n to 2 by -1 do t:= t+l[j]; r:=r, t od;
      seq([r][1+n-j], j=1..n)
    end:
seq(T(n), n=1..14); # Alois P. Heinz, Apr 05 2012

Mathematica

Table[Plus @@ (PadRight[ #,n]& /@ IntegerPartitions[n]),{n,16}]
(* Second program: *)
T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; Table[Table[T[n, k], {k, n, 1, -1}] // Accumulate // Reverse, {n, 1, 16}] // Flatten (* Jean-François Alcover, Oct 10 2015, after Omar E. Pol *)

Formula

T(n,k) = Sum_{j=1..n} A207031(j,k). - Omar E. Pol, May 02 2012

Extensions

Better definition from Omar E. Pol, Feb 13 2012