A268482 - OEIS

%I #9 Sep 27 2018 17:56:49

%S 1,-1,8,4,-76,264,-33,1248,-9735,22080,456,-32088,440448,-2085096,

%T 3715440,-9460,1216600,-26297700,205444800,-704121000,1087450320,

%U 274800,-64995600,2073673920,-23974142160,129203087760,-354403429920,500558083200

%N Triangle that arise in the study of Galois polynomials.

%H 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 First few rows are:

%e 1;

%e -1, 8;

%e 4, -76, 264;

%e -33, 1248, -9735, 22080;

%e 456, -32088, 440448, -2085096, 3715440;

%e ...

%t c[k_] := c[k] = 1 - Sum[Binomial[k, j] Binomial[k-1, j-1] c[j], {j, k-1}];

%t eul[n_, x_] := Sum[(-1)^j Binomial[n+1, j] (x-j+1)^n, {j, 0, x+1}];

%t 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 Table[(2n-1)! G[n, k], {n, 7}, {k, n}] // Flatten (* _Jean-François Alcover_, Sep 27 2018, from PARI *)

%o (PARI) C(k) = {my(j); 1 - sum(j=1, k-1, binomial(k, j)*binomial(k-1, j-1)*C(j))};

%o eul(n, x) = {my(j); sum(j=0, x+1, (-1)^j*binomial(n+1, j)*(x+1-j)^n)};

%o 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 tabl(nn) = for (n=1, nn, for (k=1, n, print1((2*n-1)!*G(n,k), ", "));print(););

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

%K sign,tabl

%O 1,3

%A _Michel Marcus_, Feb 05 2016