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A102587 T(n, k) = (-1)^n*2*[x^k] ChebyshevT(n, (1 - x)/2) with T(0,0) = 1, for 0 <= k <= n, triangle read by rows.

%I A102587 #31 Mar 24 2025 06:47:11
%S A102587 1,-1,1,-1,-2,1,2,0,-3,1,-1,4,2,-4,1,-1,-5,5,5,-5,1,2,0,-12,4,9,-6,1,
%T A102587 -1,7,7,-21,0,14,-7,1,-1,-8,12,24,-30,-8,20,-8,1,2,0,-27,9,54,-36,-21,
%U A102587 27,-9,1,-1,10,15,-60,-15,98,-35,-40,35,-10,1,-1,-11,22,66,-99,-77,154,-22,-66,44,-11,1,2,0,-48,16,180,-120,-196,216,9
%N A102587 T(n, k) = (-1)^n*2*[x^k] ChebyshevT(n, (1 - x)/2) with T(0,0) = 1, for 0 <= k <= n, triangle read by rows.
%C A102587 Previous name: Triangular matrix, read by rows, equal to the matrix inverse of triangle A094531, which is the right-hand side of trinomial table A027907.
%C A102587 Riordan array ((1-x^2)/(1+x+x^2),x/(1+x+x^2)). - _Paul Barry_, Jul 14 2005
%C A102587 Inverse of A094531. Rows sums are 1,0,-2,0,2,0,-2,... with g.f. (1-x^2)/(1+x^2). Diagonal sums are (-1)^n*C(1,n) with g.f. 1-x. - _Paul Barry_, Jul 14 2005
%C A102587 Row sums form the period 4 sequence: {1, 0,-2,0,2, 0,-2,0,2, ...}. Absolute row sums form A102588.
%C A102587 Sum_{k=0..n} T(n,k)^2 = 2*A002426(n) for n>0.
%H A102587 Feihu Liu, Ying Wang, Yingrui Zhang, and Zihao Zhang,<a href="https://arxiv.org/abs/2503.17187">Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results</a>, arXiv:2503.17187 [math.CO], 2025. See p. 10.
%H A102587 P. Peart and W.-J. Woan, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
%F A102587 T(n,k) = T(n-1,k-1) - T(n-1,k) - T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = T(2,0) = -1, T(2,1) = -2, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 22 2014
%F A102587 From _Peter Bala_, Jun 29 2015: (Start)
%F A102587 Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = x/(1 + x + x^2) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan).
%F A102587 T(n,k) = [x^(n-k)] f(x)^n with f(x) = ( 1 - x + sqrt(1 - 2*x - 3*x^2) )/2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
%e A102587 Rows begin:
%e A102587   [1],
%e A102587   [ -1,1],
%e A102587   [ -1,-2,1],
%e A102587   [2,0,-3,1],
%e A102587   [ -1,4,2,-4,1],
%e A102587   [ -1,-5,5,5,-5,1],
%e A102587   [2,0,-12,4,9,-6,1],
%e A102587   [ -1,7,7,-21,0,14,-7,1],
%e A102587   [ -1,-8,12,24,-30,-8,20,-8,1],
%e A102587   [2,0,-27,9,54,-36,-21,27,-9,1],
%e A102587   [ -1,10,15,-60,-15,98,-35,-40,35,-10,1],
%e A102587   [ -1,-11,22,66,-99,-77,154,-22,-66,44,-11,1],
%e A102587   ...
%t A102587 Table[If[n==0, 1, CoefficientList[(-1)^n 2 ChebyshevT[n, (1-x)/2], x]], {n, 0, 9}] // Flatten (* _Peter Luschny_, Mar 07 2018 *)
%o A102587 (PARI) {T(n,k)=local(A); A=matrix(n+1,n+1,r,c,if(r<c-1,0,polcoeff((1+x+x^2)^(r-1),r+c-2))); return((A^-1)[n+1,k+1])}
%o A102587 (PARI) tabl(nn) = {my(m = matrix(nn, nn, n, k, n--; k--; sum(j=0, n, binomial(n,j)*binomial(j,n-k-j)))^(-1)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ _Michel Marcus_, Jun 30 2015
%Y A102587 Cf. A094531 (matrix inverse), A102588, A002426.
%K A102587 sign,tabl
%O A102587 0,5
%A A102587 _Paul D. Hanna_, Jan 22 2005
%E A102587 New name by _Peter Luschny_, Mar 07 2018