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A355635 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}.

%I A355635 #14 Jul 26 2022 13:42:14
%S A355635 1,1,-1,1,-2,1,1,-4,5,-2,1,-8,22,-24,9,1,-16,93,-238,256,-96,1,-32,
%T A355635 386,-2180,5825,-6500,2500,1,-64,1586,-19184,117561,-345600,407700,
%U A355635 -162000,1,-128,6476,-164864,2229206,-15585920,51583084,-64538880,26471025
%N A355635 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 A355635 Without signs the triangle of elementary symmetric functions of the terms binomial(n,j), j=0..n.
%F A355635 T(n, 0) = 1.
%F A355635 T(n, 1) = -2^(n-1), for n > 0.
%F A355635 T(n, 2) = A000346(n-2), for n > 1.
%F A355635 T(n, 3) = -A025131(n-1), for n > 1.
%F A355635 T(n, 4) = A025133(n-1), for n > 1.
%F A355635 T(n, n) = (-1)^n*A001142(n-1), for n > 0.
%F A355635 T(n+1, n) = (-1)^n*A025134(n).
%F A355635 T(n+2, n) = (-1)^n*A025135(n).
%e A355635 The triangle begins:
%e A355635   1;
%e A355635   1,  -1;
%e A355635   1,  -2,   1;
%e A355635   1,  -4,   5,    -2;
%e A355635   1,  -8,  22,   -24,    9;
%e A355635   1, -16,  93,  -238,  256,   -96;
%e A355635   1, -32, 386, -2180, 5825, -6500, 2500;
%e A355635   ...
%e A355635 Row 4: x^4 - 8*x^3 + 22*x^2 - 24*x + 9 = (x-1)*(x-4)*(x-6)*(x-4)*(x-1).
%o A355635 (PARI) T(n, k) = polcoeff(prod(m=0, n, (x-binomial(n-1, m))), n-k+1);
%Y A355635 Cf. A000079, A000346, A025131, A025133, A025134, A025135.
%Y A355635 Cf. A001142 (right diagonal unsigned).
%K A355635 sign,tabl
%O A355635 0,5
%A A355635 _Thomas Scheuerle_, Jul 11 2022