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A268482 Triangle that arise in the study of Galois polynomials.

%I A268482 #9 Sep 27 2018 17:56:49
%S A268482 1,-1,8,4,-76,264,-33,1248,-9735,22080,456,-32088,440448,-2085096,
%T A268482 3715440,-9460,1216600,-26297700,205444800,-704121000,1087450320,
%U A268482 274800,-64995600,2073673920,-23974142160,129203087760,-354403429920,500558083200
%N A268482 Triangle that arise in the study of Galois polynomials.
%H A268482 Christian Günther, Kai-Uwe Schmidt, <a href="http://arxiv.org/abs/1602.01750">Lq norms of Fekete and related polynomials</a>, arXiv:1602.01750 [math.NT], 2016.
%e A268482 First few rows are:
%e A268482 1;
%e A268482 -1, 8;
%e A268482 4, -76, 264;
%e A268482 -33, 1248, -9735, 22080;
%e A268482 456, -32088, 440448, -2085096, 3715440;
%e A268482 ...
%t A268482 c[k_] := c[k] = 1 - Sum[Binomial[k, j] Binomial[k-1, j-1] c[j], {j, k-1}];
%t A268482 eul[n_, x_] := Sum[(-1)^j Binomial[n+1, j] (x-j+1)^n, {j, 0, x+1}];
%t A268482 G[k_, m_] := G[k, m] = If [k==0 && m==0, 1, Sum[Binomial[k, j] Binomial[ k-1, j-1] c[j] Sum[eul[2j-1, i-1] G[k-j, m-i], {i, m}]/(2j-1)!, {j, k}]];
%t A268482 Table[(2n-1)! G[n, k], {n, 7}, {k, n}] // Flatten (* _Jean-François Alcover_, Sep 27 2018, from PARI *)
%o A268482 (PARI) C(k) = {my(j); 1 - sum(j=1, k-1, binomial(k, j)*binomial(k-1, j-1)*C(j))};
%o A268482 eul(n, x) = {my(j); sum(j=0, x+1, (-1)^j*binomial(n+1, j)*(x+1-j)^n)};
%o A268482 G(k, m) = if ((k==0) && (m==0), 1, sum(j=1, k, binomial(k,j)*binomial(k-1,j-1)*C(j)*sum(i=1, m, eul(2*j-1,i-1)*G(k-j, m-i))/(2*j-1)!));
%o A268482 tabl(nn) = for (n=1, nn, for (k=1, n, print1((2*n-1)!*G(n,k), ", "));print(););
%Y A268482 Cf. A008292 (Eulerian numbers), A002190 (first column unsigned).
%K A268482 sign,tabl
%O A268482 1,3
%A A268482 _Michel Marcus_, Feb 05 2016