OFFSET
1,2
COMMENTS
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
.
...
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 *)
KEYWORD
AUTHOR
Wouter Meeussen, Oct 09 2010
EXTENSIONS
Better definition from Omar E. Pol, Feb 13 2012
STATUS
approved