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A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

%I A320508 #21 Dec 23 2018 23:30:54
%S A320508 1,1,-1,1,0,1,1,1,0,-1,1,2,0,0,1,1,3,1,0,0,-1,1,4,3,0,0,0,1,1,5,6,1,0,
%T A320508 0,0,-1,1,6,10,4,0,0,0,0,1,1,7,15,10,1,0,0,0,0,-1,1,8,21,20,5,0,0,0,0,
%U A320508 0,1,1,9,28,35,15,1,0,0,0,0,0,-1,1,10,36
%N A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.
%C A320508 Differs from A164925 in signs.
%C A320508 The n-th row consists of the coefficients in the expansion of (-x)^n + (((1 + sqrt(1 + 4*x))/2)^n -((1 - sqrt(1 + 4*x))/2)^n )/sqrt(1 + 4*x).
%C A320508 The coefficients in the expansion of Sum_{j=0..floor((n - 1)/2)} T(n,k)*x^(n - 2*j - 1) yield the n-th row in A168561, the coefficients of the n-th Fibonacci polynomial.
%C A320508 Row n sums up to Fibonacci(n) + (-1)^n (A008346).
%H A320508 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>
%F A320508 G.f.: 1/((1 + x*y)*(1 - y - x*y^2)).
%F A320508 E.g.f.: exp(-x*y) + (exp(y*(1 + sqrt(1 + 4*x))/2) - exp(y*(1 - sqrt(1 + 4*x))/2))/sqrt(1 + 4*x).
%F A320508 T(n,1) = A023443(n).
%e A320508 Triangle begins:
%e A320508     1;
%e A320508     1, -1;
%e A320508     1,  0,  1;
%e A320508     1,  1,  0, -1;
%e A320508     1,  2,  0,  0,  1;
%e A320508     1,  3,  1,  0,  0, -1;
%e A320508     1,  4,  3,  0,  0,  0, 1;
%e A320508     1,  5,  6,  1,  0,  0, 0, -1;
%e A320508     1,  6, 10,  4,  0,  0, 0,  0, 1;
%e A320508     1,  7, 15, 10,  1,  0, 0,  0, 0, -1;
%e A320508     1,  8, 21, 20,  5,  0, 0,  0, 0,  0, 1;
%e A320508     1,  9, 28, 35, 15,  1, 0,  0, 0,  0, 0, -1;
%e A320508     ...
%t A320508 Table[Table[Binomial[n - k - 1, k], {k, 0, n}], {n, 0, 12}]//Flatten
%o A320508 (Maxima) create_list(binomial(n - k - 1, k), n, 0, 12, k, 0, n);
%Y A320508 Inspired by A123018.
%Y A320508 Cf. A007318, A026729, A049310, A052553, A164925, A168561.
%K A320508 sign,easy,tabl
%O A320508 0,12
%A A320508 _Franck Maminirina Ramaharo_, Oct 14 2018