OFFSET
0,5
COMMENTS
Matrix inverse is triangle A104509 and is related to Fibonacci numbers. Column 0 equals A098331, with g.f.: 1/sqrt(1-2*x+5*x^2). Column 1 equals A104506, with g.f.: ((1-x)/sqrt(1-2*x+5*x^2)-1)/(2*x). Row sums equal A104507. Absolute row sums equal A104508.
Array (1/sqrt(1-2x+5x^2), (1-x-sqrt(1-2x+5x^2))/(2x)), in Riordan array notation. Product of A120616 by A007318. - Paul Barry, Jun 17 2006
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
Column k has e.g.f. exp(x)*Bessel_I(k,2*sqrt(-1)x)*(sqrt(-1))^k. - Paul Barry, Jun 17 2006
From Peter Bala, Jun 29 2015: (Start)
Matrix factorization in the Riordan group: ( 1/(1 - x), x/(1 - x) ) * ( 1/sqrt(1 + 4*x^2), (1 - sqrt(1 + 4*x^2))/(2*x) ) = A007318 * signed version of A108044.
Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x - sqrt(1 - 2*x + 5*x^2))/(2*x) 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) = x^2 + x - 1. 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;
-5,0,3,-1;
-5,8,2,-4,1;
11,15,-10,-5,5,-1;
41,-6,-30,10,9,-6,1;
29,-77,-14,49,-7,-14,7,-1;
-125,-120,112,56,-70,0,20,-8,1;
-365,117,288,-126,-126,90,12,-27,9,-1;
-131,770,45,-540,90,228,-105,-30,35,-10,1; ...
MATHEMATICA
T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, polcoeff((1+x-x^2)^n, n+k, x))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Mar 11 2005
STATUS
approved