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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.

%I A303700 #30 Sep 20 2022 07:50:04
%S A303700 1,2,-3,3,-12,10,4,-30,60,-35,5,-60,210,-280,126,6,-105,560,-1260,
%T A303700 1260,-462,7,-168,1260,-4200,6930,-5544,1716,8,-252,2520,-11550,27720,
%U A303700 -36036,24024,-6435,9,-360,4620,-27720,90090,-168168,180180,-102960,24310
%N 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.
%H A303700 Seiichi Manyama, <a href="/A303700/b303700.txt">Rows n = 0..139, flattened</a>
%F A303700 f_n(x)/(n+1) = 1/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.
%F A303700 T(n,k) = (-1)^(k)*(n+k+1)!*(k+1)/((k+1)!^2*(n-k)!). - _Jacob Fauman_, Sep 20 2022
%e A303700 Triangle begins:
%e A303700 n | 0     1     2      3     4      5     6
%e A303700 --*-----------------------------------------
%e A303700 0 | 1;
%e A303700 1 | 2,   -3;
%e A303700 2 | 3,  -12,   10;
%e A303700 3 | 4,  -30,   60,   -35;
%e A303700 4 | 5,  -60,  210,  -280,  126;
%e A303700 5 | 6, -105,  560, -1260, 1260,  -462;
%e A303700 6 | 7, -168, 1260, -4200, 6930, -5544, 1716;
%Y A303700 Cf. A253283, A303699.
%K A303700 sign,tabl
%O A303700 0,2
%A A303700 _Seiichi Manyama_, Apr 28 2018