A303699 - OEIS

%I #26 Apr 29 2018 12:48:12

%S 1,4,-6,9,-36,30,16,-120,240,-140,25,-300,1050,-1400,630,36,-630,3360,

%T -7560,7560,-2772,49,-1176,8820,-29400,48510,-38808,12012,64,-2016,

%U 20160,-92400,221760,-288288,192192,-51480,81,-3240,41580,-249480,810810,-1513512,1621620,-926640,218790

%N 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.

%H Seiichi Manyama, <a href="/A303699/b303699.txt">Rows n = 0..139, flattened</a>

%F f_n(x) = -1/n! * d^{n+1}/dx^{n+1} x^n*(1-x)^{n+1}.

%F Also f_n(x) = (n+1)/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.

%e Integral_{x=0..1} g(t - x) * (4-6*x) dx = g(t) for any polynomial g(x) of degree less than 1.

%e Triangle begins:

%e n |  0      1     2       3      4       5      6

%e --*-----------------------------------------------

%e 0 |  1;

%e 1 |  4,    -6;

%e 2 |  9,   -36,   30;

%e 3 | 16,  -120,  240,   -140;

%e 4 | 25,  -300, 1050,  -1400,   630;

%e 5 | 36,  -630, 3360,  -7560,  7560,  -2772;

%e 6 | 49, -1176, 8820, -29400, 48510, -38808, 12012;

%Y Cf. A303700.

%K sign,tabl

%O 0,2

%A _Seiichi Manyama_, Apr 28 2018