%I #14 Jul 26 2022 13:42:14

%S 1,1,-1,1,-2,1,1,-4,5,-2,1,-8,22,-24,9,1,-16,93,-238,256,-96,1,-32,

%T 386,-2180,5825,-6500,2500,1,-64,1586,-19184,117561,-345600,407700,

%U -162000,1,-128,6476,-164864,2229206,-15585920,51583084,-64538880,26471025

%N Triangle read by rows. Row n gives the coefficients of Product_{k=0..n-1} (x - binomial(n-1,k)) expanded in decreasing powers of x, with row 0 = {1}.

%C Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.

%F T(n, 0) = 1.

%F T(n, 1) = -2^(n-1), for n > 0.

%F T(n, 2) = A000346(n-2), for n > 1.

%F T(n, 3) = -A025131(n-1), for n > 1.

%F T(n, 4) = A025133(n-1), for n > 1.

%F T(n, n) = (-1)^n*A001142(n-1), for n > 0.

%F T(n+1, n) = (-1)^n*A025134(n).

%F T(n+2, n) = (-1)^n*A025135(n).

%e The triangle begins:

%e 1;

%e 1, -1;

%e 1, -2, 1;

%e 1, -4, 5, -2;

%e 1, -8, 22, -24, 9;

%e 1, -16, 93, -238, 256, -96;

%e 1, -32, 386, -2180, 5825, -6500, 2500;

%e ...

%e Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).

%o (PARI) T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);

%Y Cf. A000079, A000346, A025131, A025133, A025134, A025135.

%Y Cf. A001142 (right diagonal unsigned).

%K sign,tabl

%O 0,5

%A _Thomas Scheuerle_, Jul 11 2022