A102051 Matrix inverse of triangle A101275 (number of Schröder paths).

1, -1, 1, 3, -4, 1, -9, 15, -7, 1, 31, -58, 36, -10, 1, -113, 229, -170, 66, -13, 1, 431, -924, 775, -372, 105, -16, 1, -1697, 3795, -3481, 1939, -691, 153, -19, 1, 6847, -15822, 15542, -9674, 4072, -1154, 210, -22, 1, -28161, 66801, -69276, 47012, -22446, 7606, -1788, 276, -25, 1

Offset: 0

Author

Paul D. Hanna, Dec 27 2004

Keywords

• sign
• tabl

Comments

Row sums are {1,0,0,0...}. Absolute row sums form A006139. Column 0 forms signed A052709. Column 1 forms A102052. Column 2 forms A102053.

Examples

			Rows begin:
[1],
[ -1,1],
[3,-4,1],
[ -9,15,-7,1],
[31,-58,36,-10,1],
[ -113,229,-170,66,-13,1],
[431,-924,775,-372,105,-16,1],
[ -1697,3795,-3481,1939,-691,153,-19,1],
[6847,-15822,15542,-9674,4072,-1154,210,-22,1],...
Matrix inverse equals triangle A101275:
[1],
[1,1],
[1,4,1],
[1,13,7,1],
[1,44,34,10,1],...
		

Crossrefs

Cf. A101275, A006139, A052709, A102052, A102053, A039599.

Programs

Maxima

T(n,m):=(-1)^(n-m)*(2*m+1)*(sum((binomial(k,n-k)*binomial(2*k,k-m))/(m+k+1),k,0,n)); /* Vladimir Kruchinin, Apr 18 2015 */

PARI

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

Formula

G.f.: 2/(1+y+(1-y)*sqrt(1+4*x-4*x^2)).

T(n,m) = (-1)^(n-m)*(2*m+1)*Sum_{k=0..n} C(k,n-k)*C(2*k,k-m)/(m+k+1). - Vladimir Kruchinin, Apr 18 2015