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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.

%I A303699 #26 Apr 29 2018 12:48:12
%S A303699 1,4,-6,9,-36,30,16,-120,240,-140,25,-300,1050,-1400,630,36,-630,3360,
%T A303699 -7560,7560,-2772,49,-1176,8820,-29400,48510,-38808,12012,64,-2016,
%U A303699 20160,-92400,221760,-288288,192192,-51480,81,-3240,41580,-249480,810810,-1513512,1621620,-926640,218790
%N 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.
%H A303699 Seiichi Manyama, <a href="/A303699/b303699.txt">Rows n = 0..139, flattened</a>
%F A303699 f_n(x) = -1/n! * d^{n+1}/dx^{n+1} x^n*(1-x)^{n+1}.
%F A303699 Also f_n(x) = (n+1)/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.
%e A303699 Integral_{x=0..1} g(t - x) * (4-6*x) dx = g(t) for any polynomial g(x) of degree less than 1.
%e A303699 Triangle begins:
%e A303699 n |  0      1     2       3      4       5      6
%e A303699 --*-----------------------------------------------
%e A303699 0 |  1;
%e A303699 1 |  4,    -6;
%e A303699 2 |  9,   -36,   30;
%e A303699 3 | 16,  -120,  240,   -140;
%e A303699 4 | 25,  -300, 1050,  -1400,   630;
%e A303699 5 | 36,  -630, 3360,  -7560,  7560,  -2772;
%e A303699 6 | 49, -1176, 8820, -29400, 48510, -38808, 12012;
%Y A303699 Cf. A303700.
%K A303699 sign,tabl
%O A303699 0,2
%A A303699 _Seiichi Manyama_, Apr 28 2018