A280040 - OEIS

%I #25 Sep 06 2018 09:09:22

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

%T -32088,440448,-2085096,3715440,-2085096,440448,-32088,456,-9460,

%U 1216600,-26297700,205444800,-704121000,1087450320,-704121000,205444800,-26297700,1216600,-9460

%N Irregular triangle read by rows: numbers (2n-1)!*G(n,m) related to Galois polynomials.

%H Lars Blomberg, <a href="/A280040/b280040.txt">Table of n, a(n) for n = 1..625</a> (The first 25 rows)

%H Christian Günther, Kai-Uwe Schmidt, <a href="http://arxiv.org/abs/1602.01750">L^q norms of Fekete and related polynomials</a>, arXiv:1602.01750 [math.NT], 2016. See Corollary 2.4 p. 5.

%e Initial rows are:

%e     1,

%e    -1,    8,    -1,

%e     4,  -76,   264,   -76,     4,

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

%e   ...

%t (* "gen" stands for "generalized Eulerian number" *)

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

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

%t G[0, 0] = 1; G[k_, m_] /; 1 <= m <= 2 k - 1 := G[k, m] = Sum[Binomial[k, j] Binomial[k - 1, j - 1] c[j]/(2 j - 1)! Sum[gen[2 j - 1, i - 1] G[k - j, m - i], {i, 0, m}], {j, 1, k}]; G[_, _] = 0;

%t Table[(2 k - 1)! G[k, m], {k, 1, 6}, {m, 1, 2 k - 1}] // Flatten (* _Jean-François Alcover_, Sep 06 2018 *)

%Y Cf. A280036, A280037.

%K sign,tabf

%O 1,3

%A _N. J. A. Sloane_, Dec 28 2016

%E a(17) and beyond from _Lars Blomberg_, Jul 07 2017