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%I #6 Nov 23 2018 21:14:05
%S 1,-1,0,1,1,1,0,0,-1,-1,0,2,3,1,-1,0,0,0,0,1,1,0,0,0,1,0,1,0,0,-2,-1,
%T -2,-1,0,6,3,8,6,1,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,-1,0,-1,0,0,0,0,2,1,
%U 2,1,0,0,0,2,2,1,0,1,0,0,-6,-6,-5,-3,-3,-1,0
%N Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_polynomial">Symmetric polynomial</a>
%e Tetrangle begins (zeros not shown):
%e (1): 1
%e .
%e (2): -1
%e (11): 1 1
%e .
%e (3): 1
%e (21): -1 -1
%e (111): 2 3 1
%e .
%e (4): -1
%e (22): 1 1
%e (31): 1 1
%e (211): -2 -1 -2 -1
%e (1111): 6 3 8 6 1
%e .
%e (5): 1
%e (41): -1 -1
%e (32): -1 -1
%e (221): 2 1 2 1
%e (311): 2 2 1 1
%e (2111): -6 -6 -5 -3 -3 -1
%e (11111): 24 30 20 15 20 10 1
%e For example, row 14 gives: F(32) = -p(5) - p(32).
%Y Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.
%Y Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.
%K sign,tabf
%O 1,12
%A _Gus Wiseman_, Nov 23 2018
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