A104402 Matrix inverse of triangle A091491, read by rows.

1, -1, 1, 1, -2, 1, 0, 2, -3, 1, 0, -1, 4, -4, 1, 0, 0, -3, 7, -5, 1, 0, 0, 1, -7, 11, -6, 1, 0, 0, 0, 4, -14, 16, -7, 1, 0, 0, 0, -1, 11, -25, 22, -8, 1, 0, 0, 0, 0, -5, 25, -41, 29, -9, 1, 0, 0, 0, 0, 1, -16, 50, -63, 37, -10, 1, 0, 0, 0, 0, 0, 6, -41, 91, -92, 46, -11, 1, 0, 0, 0, 0, 0, -1, 22, -91, 154, -129, 56, -12, 1

Offset: 0

Author

Paul D. Hanna, Mar 05 2005

Keywords

• sign
• tabl

Comments

Row sums are all 0's for n>0. Absolute row sums form 2*A000045(n+1) for n>0, where A000045 = Fibonacci numbers. Sums of squared terms in row n = 2*A003440(n) for n>0, where A003440 = number of binary vectors with restricted repetitions.

Riordan array (1-x+x^2, x(1-x)). - Philippe Deléham, Nov 04 2009

Examples

			Triangle begins as:
   1;
  -1,  1;
   1, -2,  1;
   0,  2, -3,  1;
   0, -1,  4, -4,   1;
   0,  0, -3,  7,  -5,   1;
   0,  0,  1, -7,  11,  -6,  1;
   0,  0,  0,  4, -14,  16, -7,  1;
   0,  0,  0, -1,  11, -25, 22, -8, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000045, A003440, A091491, A199881.

Programs

Mathematica

Table[(-1)^(n-k)*(Binomial[k, n-k] + Binomial[k+1, n-k-1]), {n,0,12}, {k,0,n}] //Flatten (* G. C. Greubel, Apr 30 2021 *)

PARI

T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1-X+X^2)/(1-X*Y*(1-X)),n,x),k,y)

PARI

T(n,k)=if(n
				

PARI

T(n,k)=(-1)^(n-k)*(binomial(k,n-k)+binomial(k+1,n-k-1))

Sage

def A104402(n,k): return (-1)^(n+k)*(binomial(k,n-k) + binomial(k+1,n-k-1))
flatten([[A104402(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

Formula

G.f.: (1-x+x^2)/(1-x*y*(1-x)).

T(n, k) = T(n-1, k-1) - T(n-2, k-1) for k>0 with T(0, 0)=1, T(1, 0)=-1, T(2, 0)=1, T(n, 0)=0 (n>2).

T(n, k) = (-1)^(n-k)*(C(k, n-k) + C(k+1, n-k-1)).

From Philippe Deléham, Nov 04 2009: (Start)

Sum_{k=0..n} T(n,k) = 0^n.

Sum_{k=0..n} abs(T(n, k)) = 2*Fibonacci(n+1) - [n=0].

Sum_{k=0..n} ( T(n,k) )^2 = 2*A003440(n) - [n=0]. (End)