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A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*Sum_{i=0..n} x^i*binomial(n, i)/(n+i+1).

%I A327809 #10 Sep 26 2019 08:07:22
%S A327809 -1,3,2,-10,-15,-6,35,84,70,20,-126,-420,-540,-315,-70,462,1980,3465,
%T A327809 3080,1386,252,-1716,-9009,-20020,-24024,-16380,-6006,-924,6435,40040,
%U A327809 108108,163800,150150,83160,25740,3432,-24310,-175032,-556920,-1021020,-1178100,-875160,-408408,-109395,-12870
%N A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*Sum_{i=0..n} x^i*binomial(n, i)/(n+i+1).
%H A327809 Karl Dilcher, Maciej Ulas, <a href="https://arxiv.org/abs/1909.11222">Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1)=1</a>, arXiv:1909.11222 [math.NT], 2019. See Pn(x) Table 1 p. 2.
%e A327809 Triangle begins:
%e A327809      -1;
%e A327809       3,     2;
%e A327809     -10,   -15,     -6;
%e A327809      35,    84,     70,     20;
%e A327809    -126,  -420,   -540,   -315,   -70;
%e A327809     462,  1980,   3465,   3080,   1386,   252;
%e A327809   -1716, -9009, -20020, -24024, -16380, -6006, -924;
%e A327809   ...
%o A327809 (PARI) pol(n) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*sum(i=0, n, x^i*binomial(n, i)/(n+i+1));
%o A327809 row(n) = Vecrev(pol(n));
%o A327809 tabl(nn) = for (n=0, nn, print(row(n)));
%Y A327809 Cf. A046899 (Q(x) polynomials, up to sign).
%Y A327809 Cf. A001700 (1st column, up to sign), A033876 (right diagonal, up to sign).
%K A327809 sign,tabl
%O A327809 0,2
%A A327809 _Michel Marcus_, Sep 26 2019