A181472 Riordan array ((1+x)/(1+2x+2x^2),x(1+x)/(1+2x+2x^2)).

1, -1, 1, 0, -2, 1, 2, 1, -3, 1, -4, 4, 3, -4, 1, 4, -12, 5, 6, -5, 1, 0, 16, -24, 4, 10, -6, 1, -8, -4, 42, -39, 0, 15, -7, 1, 16, -32, -24, 88, -55, -8, 21, -8, 1, -16, 80, -72, -80, 159, -69, -21, 28, -9, 1, 0, -96, 240, -112, -200, 258, -77, -40, 36, -10, 1

Offset: 0

Author

Paul Barry, Oct 21 2010

Keywords

• easy
• sign
• tabl

Comments

Inverse is A054336. Coefficient array for Faber polynomials (of second kind) defined by f(x)=x+1-sum{(-1)^k/x^k,k>=1}.

Subtriangle of the triangle given by (0, -1, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. -Philippe Deléham, Feb 20 2013

Examples

			Triangle begins
1,
-1, 1,
0, -2, 1,
2, 1, -3, 1,
-4, 4, 3, -4, 1,
4, -12, 5, 6, -5, 1,
0, 16, -24, 4, 10, -6, 1,
-8, -4, 42, -39, 0, 15, -7, 1,
16, -32, -24, 88, -55, -8, 21, -8, 1
Production matrix is
-1, 1,
-1, -1, 1,
0, -1, -1, 1,
-1, 0, -1, -1, 1,
0, -1, 0, -1, -1, 1,
-2, 0, -1, 0, -1, -1, 1,
0, -2, 0, -1, 0, -1, -1, 1,
-5, 0, -2, 0, -1, 0, -1, -1, 1,
0, -5, 0, -2, 0, -1, 0, -1, -1, 1
-14, 0, -5, 0, -2, 0, -1, 0, -1, -1, 1
based on the aerated Catalan numbers.
Triangle (0, -1, 1, -2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
1
0, 1
0, -1, 1
0, 0, -2, 1
0, 2, 1, -3, 1
0, -4, 4, 3, -4, 1
0, 4, -12, 5, 6, -5, 1. -_Philippe Deléham_, Feb 20 2013
		

Formula

T(n,m)=sum(k=m..m,(-2)^(k-m)*binomial(k,n-k)*binomial(k-1,m-1)), n,m>0, [From Vladimir Kruchinin, Mar 09 2011]

T(n,k) = T(n-1,k-1) + T(n-2,k-1) -2*T(n-1,k) - 2*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -1, T(n,k) = 0 if k<0 or if k>n. -Philippe Deléham, Feb 20 2013

G.f.: (-1-x)/(-1-2*x-2*x^2+x*y+x^2*y). - R. J. Mathar, Aug 12 2015