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URL: https://oeis.org/A102587

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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
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
Feihu Liu, Ying Wang, Yingrui Zhang, and Zihao Zhang,Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results, arXiv:2503.17187 [math.CO], 2025. See p. 10.
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<c-1, 0, polcoeff((1+x+x^2)^(r-1), r+c-2))); return((A^-1)[n+1, k+1])}
(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
CROSSREFS
Cf. A094531 (matrix inverse), A102588, A002426.
Sequence in context: A071485 A127969 A081733 * A272608 A257460 A339471
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Jan 22 2005
EXTENSIONS
New name by Peter Luschny, Mar 07 2018
STATUS
approved