A303699 Triangle read by rows in which row n gives coefficients of polynomial f_n(x) 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, 4, -6, 9, -36, 30, 16, -120, 240, -140, 25, -300, 1050, -1400, 630, 36, -630, 3360, -7560, 7560, -2772, 49, -1176, 8820, -29400, 48510, -38808, 12012, 64, -2016, 20160, -92400, 221760, -288288, 192192, -51480, 81, -3240, 41580, -249480, 810810, -1513512, 1621620, -926640, 218790
Offset: 0
Examples
Integral_{x=0..1} g(t - x) * (4-6*x) dx = g(t) for any polynomial g(x) of degree less than 1.
Triangle begins:
n | 0 1 2 3 4 5 6
--*-----------------------------------------------
0 | 1;
1 | 4, -6;
2 | 9, -36, 30;
3 | 16, -120, 240, -140;
4 | 25, -300, 1050, -1400, 630;
5 | 36, -630, 3360, -7560, 7560, -2772;
6 | 49, -1176, 8820, -29400, 48510, -38808, 12012;
Links
- Seiichi Manyama, Rows n = 0..139, flattened
Crossrefs
Cf. A303700.
Formula
f_n(x) = -1/n! * d^{n+1}/dx^{n+1} x^n*(1-x)^{n+1}.
Also f_n(x) = (n+1)/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.