OFFSET
0,4
COMMENTS
Riordan array ( (1 + x^2/(1 - x - x^2), -x/(1 - x - x^2) ) belonging to the hitting time subgroup of the Riordan group (see Peart and Woan). - Peter Bala, Jun 29 2015
The sums of absolute values along steep diagonals in this triangle are: 1, 1, 3, 4 + |-1|, 7 + |-2|, 11 + |-6|, 18 + |-12| + 1, ... and these are the tribonacci numbers A000213 that begin with 1, 1, 1, 3. To see this, replace the y in the g.f. A(x,y) = (1 + x^2)/(1-x-x^2 + x*y) with y=-x^2, multiply by x, and add 1, to obtain the g.f. (1 - x^2)/(1-x-x^2-x^3) for A000213. - Noah Carey and Greg Dresden, Nov 02 2021
LINKS
Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0 to 141, flattened).
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.
Wikipedia, Lucas polynomials.
FORMULA
For n>=1, a(n,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-i,i) * binomial(n-2*i,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k). - Max Alekseyev, Oct 11 2021
G.f.: A(x, y) = (1 + x^2)/(1-x-x^2 + x*y).
G.f. for column k: g_k(x) = -(x^2+1)*x^k/(x^2+x-1)^(k+1). - Robert Israel, Jun 30 2015
G.f. for row n>=1 is the Lucas polynomial L_n(1-x). - Max Alekseyev, Oct 11 2021
EXAMPLE
Rows begin:
1;
1, -1;
3, -2, 1;
4, -6, 3, -1;
7, -12, 10, -4, 1;
11, -25, 25, -15, 5, -1;
18, -48, 60, -44, 21, -6, 1;
29, -91, 133, -119, 70, -28, 7, -1;
47, -168, 284, -296, 210, -104, 36, -8, 1;
76, -306, 585, -699, 576, -342, 147, -45, 9, -1; ...
MAPLE
S:= series((1 + x^2)/(1-x-x^2 + x*y), x, 20):
for n from 0 to 19 do R[n]:= coeff(S, x, n) od:
seq(seq(coeff(R[n], y, j), j=0..n), n=0..19); # Robert Israel, Jun 30 2015
MATHEMATICA
nmax = 11;
T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
M = Table[T[n, k], {n, 0, nmax}, {k, 0, nmax}] // Inverse;
Table[M[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)
PROG
(PARI) { T(n, k) = my(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1 + X^2)/(1-X-X^2 + X*Y), n, x), k, y); }
(PARI) { tabl(nn) = my(m = matrix(nn, nn, n, k, n--; k--; if((n<k)||(k<0), 0, polcoeff((1+x-x^2)^n, n+k, x)))^(-1)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", "); ); print(); ); } \\ Michel Marcus, Jun 30 2015
(PARI) { A104509(n, k) = if(n==0, k==0, (-1)^k * sum(i=0, (n-k)\2, n/(n-i) * binomial(n-k-i, i) * binomial(n-i, k) )); } \\ Max Alekseyev, Oct 11 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Mar 11 2005
STATUS
approved