A104495 Matrix inverse of triangle A099602, read by rows, where row n of A099602 equals the inverse binomial transform of column n of the triangle of trinomial coefficients (A027907).

1, -1, 1, 1, -2, 1, -1, 3, -4, 1, 1, -4, 12, -5, 1, -1, 5, -34, 17, -7, 1, 1, -6, 98, -51, 32, -8, 1, -1, 7, -294, 149, -124, 40, -10, 1, 1, -8, 919, -443, 448, -164, 61, -11, 1, -1, 9, -2974, 1362, -1576, 612, -298, 72, -13, 1, 1, -10, 9891, -4336, 5510, -2188, 1294, -370, 99, -14, 1, -1, 11, -33604, 14227, -19322, 7698

Offset: 0

Author

Paul D. Hanna, Mar 11 2005

Keywords

• sign
• tabl

Comments

Row sums are A104496. Absolute row sums form A014137 (partial sums of Catalan numbers). Column 2 is signed A014143.

Examples

			Rows begin:
1;
-1,1;
1,-2,1;
-1,3,-4,1;
1,-4,12,-5,1;
-1,5,-34,17,-7,1;
1,-6,98,-51,32,-8,1;
-1,7,-294,149,-124,40,-10,1;
1,-8,919,-443,448,-164,61,-11,1;
-1,9,-2974,1362,-1576,612,-298,72,-13,1; ...
		

Crossrefs

Cf. A099602, A027907, A000108, A104496, A014137, A014143.

Programs

PARI

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( (1+X*Y/(1+X))/(1+X-Y^2*(1-(1+4*X)^(1/2))^2/4),n,x),k,y)}

Formula

G.f.: A(x, y) = (1 + x*y/(1+x))/(1+x - x^2*y^2*Catalan(-x)^2), also G.f.: Column_k(x) = Catalan(-x)^(2*[k/2])/(1+x)^[(k+3)/2], where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).