|
%I #30 Oct 12 2017 08:35:49
%S -1,-1,1,-1,2,-1,3,3,-1,4,19,4,-1,5,80,65,5,-1,6,286,566,181,6,-1,7,
%T 945,3710,2905,455,7,-1,8,2997,20756,31781,12636,1079,8,-1,9,9294,
%U 105299,278304,218559,49754,2469,9
%N Triangle read by rows, a generalization of the Eulerian numbers based on Nielsen's generalized polylogarithm (case m = 2).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NielsenGeneralizedPolylogarithm.html">Nielsen Generalized Polylogarithm</a>.
%F Let p(n, m) = (m - 1)!*(1 - x)^n*PolyLog(-n, m, x)/x and P(n) the polynomial given by the expansion of p(n, m=2) after replacing log(1 - x) by 1. T(n, k) is the k-th coefficient of P(n). Using instead p(n, m=1) gives the Eulerian numbers A008292.
%e Triangle starts:
%e {-1}
%e {-1, 1}
%e {-1, 2}
%e {-1, 3, 3}
%e {-1, 4, 19, 4}
%e {-1, 5, 80, 65, 5}
%e {-1, 6, 286, 566, 181, 6}
%e {-1, 7, 945, 3710, 2905, 455, 7}
%e {-1, 8, 2997, 20756, 31781, 12636, 1079, 8}
%e {-1, 9, 9294, 105299, 278304, 218559, 49754, 2469, 9}
%e ...
%e For example with n = 4 we have p(n, x ) = (2-1)! * (1 - x)^n * PolyLog(-n, 2, x)/x
%e = x*(7 + 4*x) - (1 + 4*x + x^2)*log(1-x). Replacing log(1-x) by 1 reduces this to x*(7 + 4*x) - (1 + 4*x + x^2) = 3*x^2 + 3*x - 1 with coefficients [-1, 3, 3].
%t npl[n_, m_] := (m-1)! (1 - x)^n PolyLog[-n, m, x]/x;
%t A142249Row[n_] := CoefficientList[FunctionExpand[npl[n, 2]], x] /. Log[1-x] -> 1;
%t Table[A142249Row[n], {n, 1, 10}] // Flatten
%t (* Some older versions of Mathematica might use: *)
%t Flatten[Table[CoefficientList[Simplify[(1-x)^n * PolyLog[-n, 2, x] / (x*Log[1-x])], x]/.x->1-E, {n, 1, 15}]] (* _Vaclav Kotesovec_, Oct 12 2017 *)
%Y Row sums are A081047.
%Y A008292 (m=1), A142249 (m=2), A293298 (m=3 with an additional first column).
%Y Cf. A293561 (column 3), A293562 (column 4).
%K sign,tabl
%O 1,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 18 2008
%E Edited by _Peter Luschny_, Oct 11 2017
|
