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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. 7
 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, -1, 7, 7, -21, 0, 14, -7, 1, -1, -8, 12, 24, -30, -8, 20, -8, 1, 2, 0, -27, 9, 54, -36, -21, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. Riordan array ((1-x^2)/(1+x+x^2),x/(1+x+x^2)). - Paul Barry, Jul 14 2005 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 Row sums form the period 4 sequence: {1, 0,-2,0,2, 0,-2,0,2, ...}. Absolute row sums form A102588. Sum_{k=0..n} T(n,k)^2 = 2*A002426(n) for n>0. LINKS P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263. FORMULA 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 From Peter Bala, Jun 29 2015: (Start) 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). 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) EXAMPLE Rows begin: [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], [ -1,7,7,-21,0,14,-7,1], [ -1,-8,12,24,-30,-8,20,-8,1], [2,0,-27,9,54,-36,-21,27,-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],... MATHEMATICA Table[If[n==0, 1, CoefficientList[(-1)^n 2 ChebyshevT[n, (1-x)/2], x]], {n, 0, 9}] // Flatten (* Peter Luschny, Mar 07 2018 *) PROG (PARI) {T(n, k)=local(A); A=matrix(n+1, n+1, r, c, if(r

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Last modified October 22 19:53 EDT 2019. Contains 328319 sequences. (Running on oeis4.)