A338035 Triangle T(n,m) = (1/m)*Sum_{k=1..m} k*C(2*m-k-1,m-k)*C(2*(2*m-k),n-2*m+k), n>0, m>0.
1, 2, 1, 1, 5, 1, 0, 12, 8, 1, 0, 19, 33, 11, 1, 0, 21, 96, 63, 14, 1, 0, 15, 217, 256, 102, 17, 1, 0, 6, 386, 830, 524, 150, 20, 1, 0, 1, 533, 2241, 2147, 927, 207, 23, 1, 0, 0, 560, 5079, 7440, 4541, 1492, 273, 26, 1
Offset: 1
Examples
1, 2,1, 1,5,1, 0,12,8,1, 0,19,33,11,1, 0,21,96,63,14,1, 0,15,217,256,102,17,1
Programs
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Mathematica
T[n_, m_] := Sum[k * Binomial[2*m - k - 1, m - k] * Binomial[2*(2*m - k), n - 2*m + k], {k, 1, m}] / m; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Amiram Eldar, Oct 08 2020 *) -
Maxima
T(n,m):=sum(k*binomial(2*m-k-1,m-k)*binomial(2*(2*m-k),n-2*m+k),k,1,m)/m;
Formula
G.f.: 1/(1-(1-sqrt(x*(-4*x^5-16*x^4-24*x^3-16*x^2-4*x)*y+1))/(2*x^3+4*x^2+2*x)).