A303700 Triangle read by rows in which row n gives coefficients of polynomial f_n(x)/(n+1) of degree less than n that satisfies Integral_{x=0..1} g(t - x) * f_n(x) dx = g(t) for any polynomial g(x) of degree less than n.
1, 2, -3, 3, -12, 10, 4, -30, 60, -35, 5, -60, 210, -280, 126, 6, -105, 560, -1260, 1260, -462, 7, -168, 1260, -4200, 6930, -5544, 1716, 8, -252, 2520, -11550, 27720, -36036, 24024, -6435, 9, -360, 4620, -27720, 90090, -168168, 180180, -102960, 24310
Offset: 0
Examples
Triangle begins: n | 0 1 2 3 4 5 6 --*----------------------------------------- 0 | 1; 1 | 2, -3; 2 | 3, -12, 10; 3 | 4, -30, 60, -35; 4 | 5, -60, 210, -280, 126; 5 | 6, -105, 560, -1260, 1260, -462; 6 | 7, -168, 1260, -4200, 6930, -5544, 1716;
Links
- Seiichi Manyama, Rows n = 0..139, flattened
Formula
f_n(x)/(n+1) = 1/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.
T(n,k) = (-1)^(k)*(n+k+1)!*(k+1)/((k+1)!^2*(n-k)!). - Jacob Fauman, Sep 20 2022