A268481 - OEIS

%I #10 Sep 27 2018 11:10:40

%S 1,-2,10,16,-184,456,-272,5776,-30736,55504,7936,-284288,2555008,

%T -8998016,13801600,-353792,20594432,-280444416,1567885056,-4267790592,

%U 5960135424,22368256,-2093148160,40551058432,-325702463488,1337523883008,-3059655994368,4024935613440

%N Triangle that arise in the study of Fekete 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 -2, 10;

%e 16, -184, 456;

%e -272, 5776, -30736, 55504;

%e 7936, -284288, 2555008, -8998016, 13801600;

%e ...

%t T[k_] := T[k] = 1 - Sum[Binomial[2k-1, 2j-1] T[j], {j, 1, k-1}];

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

%t F[k_, m_] := F[k, m] = If[k == 0 && m == 0, 1, Sum[Binomial[2k-1, 2j-1] T[j] Sum[eul[2j-1, i-1] F[k-j, m-i], {i, 1, m}]/(2j-1)!, {j, 1, k}]];

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

%o (PARI) T(k) = {my(j); 1 - sum(j=1, k-1, binomial(2*k-1,2*j-1)*T(j))};

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

%o F(k, m) = if ((k==0) && (m==0), 1, sum(j=1, k, binomial(2*k-1,2*j-1)*T(j)*sum(i=1, m, eul(2*j-1,i-1)*F(k-j, m-i))/(2*j-1)!));

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

%Y Cf. A000182 (first column unsigned), A008292 (Eulerian numbers).

%K sign,tabl

%O 1,2

%A _Michel Marcus_, Feb 05 2016