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

1,2

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

Links

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

Formula

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

Example

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)

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<n := T[n, k] = T[n-k, k] + PartitionsP[n-k]; T[_, _] = 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 *)

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.

Sequence in context: A134546 A152193 A055151 * A104573 A327438 A010467

Adjacent sequences: A181184 A181185 A181186 * A181188 A181189 A181190

Keyword

easy,nonn,tabl

Author

Wouter Meeussen, Oct 09 2010

Extensions

Better definition from Omar E. Pol, Feb 13 2012

Status

approved