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%I #11 Sep 17 2023 13:40:21
%S 1,1,1,0,1,1,0,1,1,1,-1,0,1,1,1,0,-1,1,1,1,1,-1,-1,-1,1,1,1,1,0,-1,-1,
%T 0,1,1,1,1,0,-1,-2,-1,0,1,1,1,1,0,1,-1,-2,0,0,1,1,1,1,0,0,0,-2,-2,0,0,
%U 1,1,1,1
%N Triangle read by rows: T(n, m) gives the sum of the weights of weighted partitions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.
%C The row sums are given in A341417.
%C One could add a row n=0 and the column (1,repeat(0)) including the empty partition with no parts, and number of parts m = 0. The weight w(0) = -1.
%C The weight from {-1, 0, +1} of a positive number n is w(n) = 0 if n is not an element of the generalized pentagonal numbers {Pent(k) = A001318(k)}_{k>=1}, and if n = Pent(k) then w(n) = (-1)^(ceiling(Pent(k)/2)+1). The sequence
%C {(n, w(n))}_{n>=1} begins: {(1,+1), (2,+1), (3,0), (4,0), (5,-1), (6,0), (7,-1), ...}. One can also use w(0) = -1. w(n) = -A010815(n), for n >= 0. For n >= 1 w(n) = A257028(n) also.
%C The weight of a partition is the product of the weights of its parts.
%C For the triangle giving the sum of the weights of weighted compositions of n with m parts from the generalized pentagonal numbers see A341418.
%F T(n, m) = Sum_{j=1..p(n,m)} w(Part(n, m, j)), where p(n, m) = A008284(n, m), and the ternary weight of the j-th partition of n with m parts Part(n,m,j), in Abramowitz-Stegun order, is defined as the product of the weights of the parts, by w(n) = -A010815(n), for n >= 1 and m = 1, 2, ..., n.
%e The triangle T(n, m) begins:
%e n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... A341417
%e ----------------------------------------------------------------------------
%e 1: 1 1
%e 2: 1 1 2
%e 3: 0 1 1 2
%e 4: 0 1 1 1 3
%e 5; -1 0 1 1 1 2
%e 6: 0 -1 1 1 1 1 3
%e 7: -1 -1 -1 1 1 1 1 1
%e 8: 0 -1 -1 0 1 1 1 1 2
%e 9: 0 -1 -2 -1 0 1 1 1 1 0
%e 10: 0 1 -1 -2 0 0 1 1 1 1 2
%e 11: 0 0 0 -2 -2 0 0 1 1 1 1 0
%e 12: 1 1 1 0 -2 -1 0 0 1 1 1 1 4
%e 13: 0 1 1 0 -1 -2 -1 0 0 1 1 1 1 2
%e 14: 0 2 2 2 0 -1 -1 -1 0 0 1 1 1 1 7
%e 15: 1 0 1 2 1 -1 -1 -1 -1 0 0 1 1 1 1 5
%e 16: 0 1 2 2 3 1 -1 0 -1 -1 0 0 1 1 1 1 10
%e 17: 0 0 0 1 2 2 0 -1 0 -1 -1 0 0 1 1 1 1 6
%e 18: 0 0 0 2 2 3 2 0 0 0 -1 -1 0 0 1 1 1 1 11
%e 19: 0 -1 -1 -1 1 2 2 1 0 0 0 -1 -1 0 0 1 1 1 1 5
%e 20: 0 -1 -1 0 1 2 3 2 1 1 0 0 -1 -1 0 0 1 1 1 1 10
%e ...
%e n = 5: (Partition; weight w) with | separating same m numbers (in Abramowitz -Stegun order):
%e (5;-1) | (1,4;0), (2,3;0) | (1^2,3;0), (1,2^2;1) | (1^3,2;1) | (1^5;1), hence row n=5 is [-1, 0, 1, 1, 1] from the sum of same m weights.
%Y Cf. A000045, A001318, A008284, -A010815, A257028, A341417, A341418.
%K sign,tabl
%O 1,39
%A _Wolfdieter Lang_, Feb 15 2021
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