A234922 Array w(n,h), in which row n shows the weights (defined in Comments) of the partitions of n, arranged in reverse Mathematica order.

1, 3, 2, 6, 5, 3, 10, 9, 6, 7, 4, 15, 14, 11, 12, 8, 9, 5, 21, 20, 17, 12, 18, 14, 9, 15, 10, 11, 6, 28, 27, 24, 19, 25, 21, 15, 16, 22, 17, 11, 18, 12, 13, 7, 36, 35, 32, 27, 20, 33, 29, 23, 24, 17, 30, 25, 18, 19, 12, 26, 20, 13, 21, 14, 15, 8, 45, 44, 41

Offset

1,2

Comments

The weight of a partition P = x(1)+ x(2)+...+x(k) of n is introduced at A234094 as k*x(1) + (k-1)*x(2) + ... + x(k).

Links

Table of n, a(n) for n=1..69.

Formula

w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).

Example

Represent 1+1+1+1+1 as _1_1_1_1_1_.  The partition 1+2+2 matches the placement of dividers d indicated by _1d1_1d1_1d. To place the 1st d takes 1 step (starting at the 1st '_'); to place the 2nd d takes 1+2 steps (starting at the 1st '_'); to place the 3rd d takes 1+2+2 steps. The total number of steps is 2+3+5 = 9, the 3rd number in row 5, because 1+2+2 is the 3rd partition of 5 in reverse Mathematica ordering. The first 6 rows:
1
3    2
6    5    3
10   9    6    7    4
15   14   11   12   8    9    5
21   20   17   12   18   14   9   15   10   11   6

Mathematica

p[n_] := p[n] = Reverse[IntegerPartitions[n]]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]];
Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]] (* A234094 *)
TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]

Crossrefs

Cf. A234094, A234097.

Sequence in context: A088452 A268719 A297878 * A049777 A193999 A210971

Adjacent sequences: A234919 A234920 A234921 * A234923 A234924 A234925

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Jan 01 2014

Status

approved