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%I #10 Jan 10 2019 02:20:44
%S -1,1,-1,5,-1,-1,25,-5,-1,-1,125,-25,-5,-1,-1,625,-125,-25,-5,-1,-1,
%T 3125,-625,-125,-25,-5,-1,-1,15625,-3125,-625,-125,-25,-5,-1,-1,78125,
%U -15625,-3125,-625,-125,-25,-5,-1,-1,390625,-78125,-15625,-3125,-625,-125,-25,-5,-1,-1
%N Triangle T(n,k) read by rows: T(n,n) = -1, T(n,0) = 5^(n - 1), T(n,k) = -5^(n - k - 1), 1 <= k <= n - 1
%F From _Franck Maminirina Ramaharo_, Jan 08 2019: (Start)
%F G.f.: -(1 - 6*y + 2*x*y^2)/(1 - (5 + x)*y + 5*x*y^2).
%F E.g.f.: -(10 - 2*x - (5 - 2*x)*exp(5*y) + (20 - 5*x)*exp(x*y))/(25 - 5*x). (End)
%e Triangle begins:
%e -1;
%e 1, -1;
%e 5, -1, -1;
%e 25, -5, -1, -1;
%e 125, -25, -5, -1, -1;
%e 625, -125, -25, -5, -1, -1;
%e 3125, -625, -125, -25, -5, -1, -1;
%e 15625, -3125, -625, -125, -25, -5, -1, -1;
%e 78125, -15625, -3125, -625, -125, -25, -5, -1, -1;
%e 390625, -78125, -15625, -3125, -625, -125, -25, -5, -1, -1;
%e 1953125, -390625, -78125, -15625, -3125, -625, -125, -25, -5, -1, -1;
%e ...
%t b[0] = {-1}; b[1] = {1, -1};
%t b[n_] := b[n] = Join[{5^(n - 1)}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
%t Flatten[Table[b[n], {n, 0, 10}]]
%o (Maxima)
%o T(n, k) := if k = n then -1 else if k = 0 then 5^(n - 1) else -5^(n - k - 1);
%o create_list(T(n, k), n, 0, 20, k, 0, n); /* _Franck Maminirina Ramaharo_, Jan 08 2019 */
%Y Row sums (except row 0): A125833.
%Y Cf. A057728, A152568, A152570, A152571.
%K sign,easy,tabl
%O 0,4
%A _Roger L. Bagula_, Dec 08 2008
%E Edited by _Franck Maminirina Ramaharo_, Jan 08 2019
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