OFFSET
1,5
LINKS
Indranil Ghosh, Rows 1..125, flattened
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
FORMULA
G.f.: -(2*x^2*y)/(2*x^2*y+sqrt(1-4*x)-1).
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.
Sum_{n>=m} T(n,m)x^m is expansion of ((2*x^2)/(1-sqrt(1-4*x)))^m.
EXAMPLE
1;
-1,1;
-1,-2,1;
-2,-1,-3,1;
-5,-2,0,-4,1;
-14,-5,-1,2,-5,1;
-42,-14,-3,0,5,-6,1;
MATHEMATICA
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 *)
PROG
(Maxima)
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);
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);
taylor(-(2*x^2*y)/(2*x^2*y+sqrt(1-4*x)-1), x, 0, 7, y, 0, 7);
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Vladimir Kruchinin, Apr 16 2016
STATUS
approved