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Triangle T(n,m) = Sum_{k=1..n-m} (k*(-1)^k*binomial(m+k-1,k)*binomial(2*(n-m),n-m-k))/(n-m), with T(n,n)=1.
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%I #14 Jul 11 2018 20:12:32

%S 1,-1,1,-1,-2,1,-2,-1,-3,1,-5,-2,0,-4,1,-14,-5,-1,2,-5,1,-42,-14,-3,0,

%T 5,-6,1,-132,-42,-9,-1,0,9,-7,1,-429,-132,-28,-4,0,-2,14,-8,1,-1430,

%U -429,-90,-14,-1,0,-7,20,-9,1,-4862,-1430,-297,-48,-5,0,0,-16,27,-10,1

%N Triangle T(n,m) = Sum_{k=1..n-m} (k*(-1)^k*binomial(m+k-1,k)*binomial(2*(n-m),n-m-k))/(n-m), with T(n,n)=1.

%H Indranil Ghosh, <a href="/A271875/b271875.txt">Rows 1..125, flattened</a>

%H Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.

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

%F T(n,m) = Sum_{k=1..n-m}((k*(-1)^k*binomial(m+k-1,k)*binomial(n,n-m-k)*binomial(2*n-2*m+k-1,n-m))/(n-m+k)), T(n,n)=1.

%F Sum_{n>=m} T(n,m)x^m is expansion of ((2*x^2)/(1-sqrt(1-4*x)))^m.

%e 1;

%e -1,1;

%e -1,-2,1;

%e -2,-1,-3,1;

%e -5,-2,0,-4,1;

%e -14,-5,-1,2,-5,1;

%e -42,-14,-3,0,5,-6,1;

%t T[n_,m_]:= T[n,m]=Sum[(k*(-1)^k*Binomial[m+k-1,k]*Binomial[2*(n-m),n-m-k])/(n-m),{k,1,n-m}];Flatten[Table[If[n==k,1,T[n,k]],{n,1,11},{k,1,n}]] (* _Indranil Ghosh_, Feb 20 2017 *)

%o (Maxima)

%o T(n,m):=if n=m then 1 else sum(k*(-1)^k*binomial(m+k-1,k)*binomial(2*(n-m),n-m-k),k,1,n-m)/(n-m);

%o T(n,m):=if n=m then 1 else sum((k*(-1)^k*binomial(m+k-1,k)*binomial(n,n-m-k)*binomial(2*n-2*m+k-1,n-m))/(n-m+k),k,1,n-m);

%o taylor(-(2*x^2*y)/(2*x^2*y+sqrt(1-4*x)-1),x,0,7,y,0,7);

%Y Cf. A000108.

%K sign,tabl

%O 1,5

%A _Vladimir Kruchinin_, Apr 16 2016