A338036 Triangle T(n,m) = Sum_{k=1..m} C(2*m-k-1,m-k)*C(2*(2*m-k),n-2*m+k), n>0, m>0.
1, 2, 1, 1, 6, 1, 0, 18, 9, 1, 0, 34, 45, 12, 1, 0, 41, 164, 78, 15, 1, 0, 30, 453, 376, 120, 18, 1, 0, 12, 936, 1490, 695, 171, 21, 1, 0, 2, 1429, 4916, 3305, 1158, 231, 24, 1, 0, 0, 1596, 13266, 13647, 6333, 1792, 300, 27, 1
Offset: 1
Examples
1, 2,1, 1,6,1, 0,18,9,1, 0,34,45,12,1, 0,41,164,78,15,1, 0,30,453,376,120,18,1
Programs
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Mathematica
T[n_, m_] := Sum[Binomial[2*m - k - 1, m - k] * Binomial[2*(2*m - k), n - 2*m + k], {k, 1, m}]; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Amiram Eldar, Oct 08 2020 *) -
Maxima
T(n,m):=sum(binomial(2*m-k-1,m-k)*binomial(2*(2*m-k),n-2*m+k),k,1,m);
Formula
G.f.: 2*x^2*(x+1)^4/(1-4*x^2*(x+1)^4*y+(2*x*(x+1)^2-1)*sqrt(1-4*x^2*(x+1)^4*y)).