OFFSET
0,1
COMMENTS
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
EXAMPLE
Triangle begins:
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, -24, 10;
-20, 70, -84, 35, 35, -84, 70, -20;
-35, 120, -140, 56, 0, 56, -140, 120, -35;
70, -315, 540, -420, 126, 126, -420, 540, -315, 70;
126, -560, 945, -720, 210, 0, 210, -720, 945, -560, 126;
...
MATHEMATICA
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]}]]];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 13 2009
STATUS
approved