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A234094 Array {w(n,h)}: row n shows the weights, as defined in Comments, of the partitions of n, arranged in Mathematica order.

%I A234094 #9 Apr 03 2014 03:14:20
%S A234094 1,2,3,3,5,6,4,7,6,9,10,5,9,8,12,11,14,15,6,11,10,15,9,14,18,12,17,20,
%T A234094 21,7,13,12,18,11,17,22,16,15,21,25,19,24,27,28,8,15,14,21,13,20,26,
%U A234094 12,19,18,25,30,17,24,23,29,33,20,27,32,35,36,9,17,16
%N A234094 Array {w(n,h)}: row n shows the weights, as defined in Comments, of the partitions of n, arranged in Mathematica order.
%C A234094 The weight of a partition P = x(1)+x(2)+...+x(k) of n is introduced here as k*x(1)+(k-1)*x(2)+...+x(k), which is the number of steps needed to make P from the sum 1+1+...+1 = n by moving dividers (or parentheses) into the sum; see the Example section.
%F A234094 w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).
%e A234094 Represent 1+1+1+1+1 as _1_1_1_1_1_.  The partition 2+2+1 matches the placement of dividers d indicated by _1_1d1_1d1d.  To place the 1st d takes 2 steps (starting at the 1st _); to place the 2nd d takes 2+2 = 4 steps (starting at the 1st _ ); to place the 3rd d takes 2+2+1 = 5 steps.  The total number of steps is 2+4+5 = 11, which is the 5th number in row 5 because 2+2+1 is the 5th partition of 5 in Mathematica ordering.  The first 6 rows are:
%e A234094 1
%e A234094 2 ... 3
%e A234094 3 ... 5 ... 6
%e A234094 4 ... 7 ... 6 ... 9 ... 10
%e A234094 5 ... 9 ... 8 ... 12 .. 11 .. 14 ... 15
%e A234094 6 ... 11 .. 10 .. 15 .. 9 ... 14 ... 18 .. 12 .. 17 .. 20 .. 21
%t A234094 p[n_] := p[n] = 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]]]]];
%t A234094 Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]] (* A234094 *)
%t A234094 TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]
%Y A234094 Cf. A234097, A234922, A234924.
%K A234094 nonn,easy,tabf
%O A234094 1,2
%A A234094 _Clark Kimberling_, Jan 01 2014