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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 29 14:23 EST 2020. Contains 331338 sequences. (Running on oeis4.)