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.
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
Examples
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], ...
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.
Programs
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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 *) -
PARI
{T(n,k)=local(A); A=matrix(n+1,n+1,r,c,if(r -
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
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)
Extensions
New name by Peter Luschny, Mar 07 2018
Comments