A268482 Triangle that arise in the study of Galois polynomials.

1, -1, 8, 4, -76, 264, -33, 1248, -9735, 22080, 456, -32088, 440448, -2085096, 3715440, -9460, 1216600, -26297700, 205444800, -704121000, 1087450320, 274800, -64995600, 2073673920, -23974142160, 129203087760, -354403429920, 500558083200

Offset: 1

Author

Michel Marcus, Feb 05 2016

Keywords

• sign
• tabl

Examples

			First few rows are:
1;
-1, 8;
4, -76, 264;
-33, 1248, -9735, 22080;
456, -32088, 440448, -2085096, 3715440;
...
		

Links

• Christian Günther, Kai-Uwe Schmidt, Lq norms of Fekete and related polynomials, arXiv:1602.01750 [math.NT], 2016.

Crossrefs

Cf. A008292 (Eulerian numbers), A002190 (first column unsigned).

Programs

Mathematica

c[k_] := c[k] = 1 - Sum[Binomial[k, j] Binomial[k-1, j-1] c[j], {j, k-1}];
eul[n_, x_] := Sum[(-1)^j Binomial[n+1, j] (x-j+1)^n, {j, 0, x+1}];
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}]];
Table[(2n-1)! G[n, k], {n, 7}, {k, n}] // Flatten (* Jean-François Alcover, Sep 27 2018, from PARI *)

PARI

C(k) = {my(j); 1 - sum(j=1, k-1, binomial(k, j)*binomial(k-1, j-1)*C(j))};
eul(n, x) = {my(j); sum(j=0, x+1, (-1)^j*binomial(n+1, j)*(x+1-j)^n)};
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)!));
tabl(nn) = for (n=1, nn, for (k=1, n, print1((2*n-1)!*G(n,k), ", "));print(););