A097808 Riordan array ((1+2x)/(1+x)^2, 1/(1+x)) read by rows.

1, 0, 1, -1, -1, 1, 2, 0, -2, 1, -3, 2, 2, -3, 1, 4, -5, 0, 5, -4, 1, -5, 9, -5, -5, 9, -5, 1, 6, -14, 14, 0, -14, 14, -6, 1, -7, 20, -28, 14, 14, -28, 20, -7, 1, 8, -27, 48, -42, 0, 42, -48, 27, -8, 1, -9, 35, -75, 90, -42, -42, 90, -75, 35, -9, 1, 10, -44, 110, -165, 132, 0, -132, 165, -110, 44, -10, 1

Offset

0,7

Comments

Inverse of A059260. Row sums are inverse binomial transform of A040000, with g.f. (1+2x)/(1+x). Diagonal sums are (-1)^n(1-Fib(n)). A097808=B^(-1)*A097806, where B is the binomial matrix. B*A097808*B^(-1) is the inverse of A097805.

Links

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)

Formula

Columns have g.f. (1+2x)/(1+x)^2(x/(1+x))^k.

T(n,k)=T(n-1,k-1)-3*T(n-1,k)+2*T(n-2,k-1)-3*T(n-2,k)+T(n-3,k-1)-T(n-3,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=0, T(2,0)=T(2,1)=-1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 12 2014

T(0,0)=1, T(n,0)=(-1)^(n-1)*(n-1) for n>0, T(n,n)=1, T(n,k)=T(n-1,k-1)-T(n-1,k) for 0<k<n. - Philippe Deléham, Jan 12 2014

Example

Rows begin
1;
0, 1;
-1, -1, 1;
2, 0, -2, 1;
-3, 2, 2, -3, 1;
4, -5, 0, 5, -4, 1;
-5, 9, -5, -5, 9, -5, 1;
6, -14, 14, 0, -14, 14, -6, 1;
-7, 20, -28, 14, 14, -28, 20, -7, 1;
8, -27, 48, -42, 0, 42, -48, 27, -8, 1;

Maple

T:= proc(n, k) option remember;
if k < 0 or k > n then return 0 fi;
procname (n-1, k-1)-3*procname(n-1, k)+2*procname(n-2, k-1)-3*procname(n-2, k)+
procname(n-3, k-1)-procname(n-3, k)
end proc:
T(0, 0):= 1: T(1, 1):= 1: T(2, 2):= 1:
T(1, 0):= 0: T(2, 0):= -1: T(2, 1):= -1:
seq(seq(T(n, k), k=0..n), n=0..12); # Robert Israel, Jul 16 2019

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 + 2 #)/(1 + #)^2&, #/(1 + #)&, 12] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Crossrefs

Sequence in context: A138110 A080233 A156644 * A349463 A114325 A101048

Adjacent sequences: A097805 A097806 A097807 * A097809 A097810 A097811

Keyword

easy,sign,tabl

Author

Paul Barry, Aug 25 2004

Status

approved