OFFSET
0,2
COMMENTS
Inverse of Riordan array A094527. Rows sums are A099837. Diagonal sums are A110164. Product of Riordan array A102587 and inverse binomial transform (1/(1+x), x/(1+x)).
Coefficients of polynomials related to Cartan matrices of types C_n and B_n: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2), with p(x,0) = 1; p(x,1) = 2-x; p(x,2) = x^2-4*x-2. - Roger L. Bagula, Apr 12 2008
From Wolfdieter Lang, Nov 16 2012: (Start)
The alternating row sums are given in A219233.
For n >= 1 the row polynomials in the variable x^2 are R(2*n,x):=2*T(2*n,x/2) with Chebyshev's T-polynomials. See A127672 and also the triangle A127677.
(End)
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)^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 - 2*x + sqrt(1 - 4*x))/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)
LINKS
G. C. Greubel, Rows n=0..100 of triangle, flattened
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
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.
T. M. Richardson, The Reciprocal Pascal Matrix, arXiv:1405.6315 [math.CO], 2014.
FORMULA
T(n,k) = (-1)^(n-k)*(C(n+k,n-k) + C(n+k-1,n-k-1)), with T(0,0) = 1. - Paul Barry, Mar 22 2007
From Wolfdieter Lang, Nov 16 2012: (Start)
O.g.f. row polynomials P(n,x) := Sum(T(n,k)*x^k, k=0..n): (1-z^2)/(1+(x-2)*z+z^2) (from the Riordan property).
O.g.f. column No. k: ((1-x)/(1+x))*(x/(1+x)^2)^k, k >= 0.
T(0,0) = 1, T(n,k) = (-1)^(n-k)*(2*n/(n+k))*binomial(n+k,n-k), n>=1, and T(n,k) = 0 if n < k. (From the Chebyshev T-polynomial formula due to Waring's formula.)
(End)
T(n,k) = -2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 29 2013
EXAMPLE
Triangle T(n,k) begins:
m\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: -2 1
2: 2 -4 1
3: -2 9 -6 1
4: 2 -16 20 -8 1
5: -2 25 -50 35 -10 1
6: 2 -36 105 -112 54 -12 1
7: -2 49 -196 294 -210 77 -14 1
8: 2 -64 336 -672 660 -352 104 -16 1
9: -2 81 -540 1386 -1782 1287 -546 135 -18 1
10: 2 -100 825 -2640 4290 -4004 2275 -800 170 -20 1
... Reformatted and extended by Wolfdieter Lang, Nov 16 2012
Row polynomial n=2: P(2,x) = 2 - 4*x + x^2. R(4,x):= 2*T(4,x/2) = 2 - 4*x^2 + x^4. For P and R see a comment above. - Wolfdieter Lang, Nov 16 2012.
MATHEMATICA
Table[If[n==0 && k==0, 1, (-1)^(n-k)*(Binomial[n+k, n-k] + Binomial[n+k-1, n-k-1])], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 16 2018 *)
PROG
(Magma) /* As triangle */ [[(-1)^(n-k)*(Binomial(n+k, n-k) + Binomial(n+k-1, n-k-1)): k in [0..n]]: n in [0.. 12]]; // Vincenzo Librandi, Jun 30 2015
(PARI) {T(n, k) = (-1)^(n-k)*(binomial(n+k, n-k) + binomial(n+k-1, n-k-1))};
for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 16 2018
(Sage) [[(-1)^(n-k)*(binomial(n+k, n-k) + binomial(n+k-1, n-k-1)) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Dec 16 2018
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 14 2005
STATUS
approved