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%I #9 Apr 22 2015 16:22:11
%S 1,-1,1,3,-4,1,-9,15,-7,1,31,-58,36,-10,1,-113,229,-170,66,-13,1,431,
%T -924,775,-372,105,-16,1,-1697,3795,-3481,1939,-691,153,-19,1,6847,
%U -15822,15542,-9674,4072,-1154,210,-22,1,-28161,66801,-69276,47012,-22446,7606,-1788,276,-25,1
%N Matrix inverse of triangle A101275 (number of Schröder paths).
%C 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.
%F G.f.: 2/(1+y+(1-y)*sqrt(1+4*x-4*x^2)).
%F 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
%e Rows begin:
%e [1],
%e [ -1,1],
%e [3,-4,1],
%e [ -9,15,-7,1],
%e [31,-58,36,-10,1],
%e [ -113,229,-170,66,-13,1],
%e [431,-924,775,-372,105,-16,1],
%e [ -1697,3795,-3481,1939,-691,153,-19,1],
%e [6847,-15822,15542,-9674,4072,-1154,210,-22,1],...
%e Matrix inverse equals triangle A101275:
%e [1],
%e [1,1],
%e [1,4,1],
%e [1,13,7,1],
%e [1,44,34,10,1],...
%o (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)}
%o (Maxima)
%o 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 */
%Y Cf. A101275, A006139, A052709, A102052, A102053, A039599.
%K sign,tabl
%O 0,4
%A _Paul D. Hanna_, Dec 27 2004
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