|
%I #4 Nov 20 2018 16:30:45
%S 1,1,-2,1,0,1,3,-3,1,0,-2,1,-4,2,4,-4,1,0,0,1,0,4,0,-4,1,0,0,3,-3,1,5,
%T -5,-5,5,5,-5,1,0,0,0,-2,1,-6,6,6,3,-2,-6,-12,9,6,-6,1,0,-4,0,2,4,-4,
%U 1,0,0,-6,6,3,-5,1,0,0,0,0,1,7,-7,-7,-7,14,7,7
%N Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of e(v) in p(u), where H is Heinz number, e is elementary symmetric functions, and p is power sum symmetric functions.
%C Row n has length A000041(A056239(n)).
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e Triangle begins:
%e 1
%e 1
%e -2 1
%e 0 1
%e 3 -3 1
%e 0 -2 1
%e -4 2 4 -4 1
%e 0 0 1
%e 0 4 0 -4 1
%e 0 0 3 -3 1
%e 5 -5 -5 5 5 -5 1
%e 0 0 0 -2 1
%e -6 6 6 3 -2 -6 -12 9 6 -6 1
%e 0 -4 0 2 4 -4 1
%e 0 0 -6 6 3 -5 1
%e 0 0 0 0 1
%e 7 -7 -7 -7 14 7 7 7 -7 -7 -21 14 7 -7 1
%e 0 0 0 4 0 -4 1
%e For example, row 15 gives: p(32) = -6e(32) + 6e(221) + 3e(311) - 5e(2111) + e(11111).
%Y Row sums are A321753.
%Y Cf. A005651, A008480, A056239, A124794, A124795, A135278, A296150, A319193, A319225, A319226, A321742-A321765, A321854.
%K sign,tabf
%O 1,3
%A _Gus Wiseman_, Nov 20 2018
|
