cp's OEIS Frontend

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

%I A102051 #9 Apr 22 2015 16:22:11
%S A102051 1,-1,1,3,-4,1,-9,15,-7,1,31,-58,36,-10,1,-113,229,-170,66,-13,1,431,
%T A102051 -924,775,-372,105,-16,1,-1697,3795,-3481,1939,-691,153,-19,1,6847,
%U A102051 -15822,15542,-9674,4072,-1154,210,-22,1,-28161,66801,-69276,47012,-22446,7606,-1788,276,-25,1
%N A102051 Matrix inverse of triangle A101275 (number of Schröder paths).
%C A102051 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 A102051 G.f.: 2/(1+y+(1-y)*sqrt(1+4*x-4*x^2)).
%F A102051 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 A102051 Rows begin:
%e A102051 [1],
%e A102051 [ -1,1],
%e A102051 [3,-4,1],
%e A102051 [ -9,15,-7,1],
%e A102051 [31,-58,36,-10,1],
%e A102051 [ -113,229,-170,66,-13,1],
%e A102051 [431,-924,775,-372,105,-16,1],
%e A102051 [ -1697,3795,-3481,1939,-691,153,-19,1],
%e A102051 [6847,-15822,15542,-9674,4072,-1154,210,-22,1],...
%e A102051 Matrix inverse equals triangle A101275:
%e A102051 [1],
%e A102051 [1,1],
%e A102051 [1,4,1],
%e A102051 [1,13,7,1],
%e A102051 [1,44,34,10,1],...
%o A102051 (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 A102051 (Maxima)
%o A102051 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 A102051 Cf. A101275, A006139, A052709, A102052, A102053, A039599.
%K A102051 sign,tabl
%O A102051 0,4
%A A102051 _Paul D. Hanna_, Dec 27 2004