%I #14 Aug 15 2020 09:41:47

%S 1,1,-1,0,-1,1,-2,4,-2,-4,14,-17,8,-1,0,4,-13,15,-7,1,8,-32,46,-25,-1,

%T 5,-1,8,-48,116,-144,96,-32,4,0,-24,132,-300,361,-244,90,-16,1,-16,96,

%U -228,252,-79,-109,134,-62,13,-1,-32,272,-984,1980,-2416,1811,-787,154,10,-9,1

%N Triangle of coefficients from polynomial recursion P(x,n)=(1-x)*P(x,n-1) - binomial(x-1,2)*P(x,n-2).

%C Row sums are 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...

%H Michael Gromov, <a href="http://www.numdam.org/article/PMIHES_1981__53__53_0.pdf">Groups of polynomial growth and expanding maps (with an appendix by Jacques Tits)</a>, Publications Math. de l'IHES, 53 (1981), p. 53-78; see p. 75

%F P[x, -1] = 0; P[x, 0] = 1; P[x, 1] = 1 - x; P(x,n)=(1-x)*P(x,n-1)-binomial[x-1,2]*P(x,n-2) Output as 2^Floor[n/2]*P(x,n) to get Integers.

%e 1;

%e 1, -1;

%e 0, -1, 1;

%e -2, 4, -2;

%e -4, 14, -17,8, -1;

%e 0, 4, -13, 15, -7, 1;

%e 8, -32, 46, -25, -1, 5, -1;

%e 8, -48, 116, -144, 96, -32, 4;

%e 0, -24, 132, -300, 361, -244,90, -16, 1;

%e -16, 96, -228, 252, -79, -109, 134, -62, 13, -1;

%e -32, 272, -984, 1980, -2416, 1811, -787, 154, 10, -9, 1;

%t P[x, -1] = 0; P[x, 0] = 1; P[x, 1] = 1 - x; P[x_, n_] := P[x, n] = (1 - x)*P[x, n - 1] - Binomial[x - 1, 2]*P[x, n - 2];

%t Table[ExpandAll[2^Floor[n/2]*P[x, n]], {n, 0, 10}];

%t a = Table[CoefficientList[2^Floor[n/2]*P[x, n], x], {n, 0, 10}]; Flatten[a]

%K uned,tabl,sign

%O 1,7

%A _Roger L. Bagula_, Mar 31 2008