|
%I #14 Oct 23 2018 11:09:42
%S 1,-1,1,-1,1,1,-1,1,2,1,-1,1,3,3,1,-1,1,4,6,4,1,-1,1,5,10,10,5,1,-1,1,
%T 6,15,20,15,6,1,-1,1,7,21,35,35,21,7,1,-1,1,8,28,56,70,56,28,8,1,-1,1,
%U 9,36,84,126,126,84,36,9,1,-1,1,10,45,120,210,252,210,120,45,10,1
%N Triangle read by rows: T(n,k) = binomial(n - 1, k - 1), 1 <= k <= n, and T(n,0) = A153881(n+1), n >= 0.
%C Row sums yield A000225 preceded by 1.
%C Except for signs, this is A135225.
%F Row n = coefficients in the expansion of x*(1 + x)^(n - 1) - 1, n > 0.
%F From _Franck Maminirina Ramaharo_, Oct 23 2018: (Start)
%F G.f.: (1 - 3*y + (2 + x)*y^2)/(1 - (2 + x)*y + (1 + x)*y^2).
%F E.g.f.: (2 + x - (1 + x)*exp(y) + x*exp((1 + x)*y))/(1 + x). (End)
%e Triangle begins:
%e 1;
%e -1, 1;
%e -1, 1, 1;
%e -1, 1, 2, 1;
%e -1, 1, 3, 3, 1;
%e -1, 1, 4, 6, 4, 1;
%e -1, 1, 5, 10, 10, 5, 1;
%e -1, 1, 6, 15, 20, 15, 6, 1;
%e -1, 1, 7, 21, 35, 35, 21, 7, 1;
%e -1, 1, 8, 28, 56, 70, 56, 28, 8, 1;
%e -1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;
%e ...
%t Flatten[Table[If[n == 0, {1}, CoefficientList[x*(1 + x)^( n - 1) - 1, x]], {n, 0, 10}]]
%o (Maxima)
%o T(n, k) := if k = 0 then 2*floor(1/(n + 1)) - 1 else binomial(n - 1, k - 1)$
%o create_list(T(n, k), n, 0, 12, k, 0, n); /* _Franck Maminirina Ramaharo_, Oct 23 2018 */
%Y Cf. A007318, A014473, A097805, A135225.
%K sign,tabl,easy
%O 0,9
%A _Roger L. Bagula_, May 13 2010
%E Edited and new name by _Franck Maminirina Ramaharo_, Oct 23 2018
|
