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%I #19 Sep 19 2022 23:33:10
%S 2,1,1,1,0,1,-2,3,3,-2,-3,4,0,4,-3,6,-15,10,10,-15,6,10,-24,15,0,15,
%T -24,10,-20,70,-84,35,35,-84,70,-20,-35,120,-140,56,0,56,-140,120,-35,
%U 70,-315,540,-420,126,126,-420,540,-315,70,126,-560,945,-720,210,0,210
%N Triangle read by rows: p(x,n) = 2 if n = 0, Sum_{i=0..floor((n-1)/2)} binomial(n, i)*(x - 1)^i + x^n*Sum_{i=0..floor((n-1)/2)} binomial(n, i)*(1/x - 1)^i otherwise.
%C The first half of every second row gives the coefficients of a polynomial approximation of f(0) = f'(0) = f'(1) = f''(0) = f''(1) = ... = 0 and f(1)=1: x, -2x^3 + 3x^2, 6x^5 - 15x^4 + 10x^3, ... - _Martin Clever_, Sep 12 2022
%e Triangle begins:
%e 2;
%e 1, 1;
%e 1, 0, 1;
%e -2, 3, 3, -2;
%e -3, 4, 0, 4, -3;
%e 6, -15, 10, 10, -15, 6;
%e 10, -24, 15, 0, 15, -24, 10;
%e -20, 70, -84, 35, 35, -84, 70, -20;
%e -35, 120, -140, 56, 0, 56, -140, 120, -35;
%e 70, -315, 540, -420, 126, 126, -420, 540, -315, 70;
%e 126, -560, 945, -720, 210, 0, 210, -720, 945, -560, 126;
%e ...
%t p[x_, n_] = If[ n == 0, 2, Sum[Binomial[ n, i]*(x - 1)^i, {i, 0, Floor[(n - 1)/2]}] + Expand[x^n*Sum[Binomial[n, i]*(1/x - 1)^ i, {i, 0, Floor[(n - 1)/2]}]]];
%t Table[CoefficientList[p[x, n], x], {n, 0, 10}];
%t Flatten[%]
%K sign,tabl,uned,less
%O 0,1
%A _Roger L. Bagula_, Mar 13 2009
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