A350557 Triangle T(n,k) read by rows with T(n,0) = (2*n)! / (2^n * n!) for n >= 0 and T(n,k) = (Sum_{i=k..n} binomial(i-1,k-1) * 2^i * i! / (2*i)!) * (2*n)! / (2^n * n!) for 0 < k <= n.
1, 1, 1, 3, 4, 1, 15, 21, 7, 1, 105, 148, 52, 10, 1, 945, 1333, 472, 96, 13, 1, 10395, 14664, 5197, 1066, 153, 16, 1, 135135, 190633, 67567, 13873, 2009, 223, 19, 1, 2027025, 2859496, 1013512, 208116, 30170, 3380, 306, 22, 1
Offset: 0
Examples
Triangle T(n,k) for 0 <= k <= n starts: n\k : 0 1 2 3 4 5 6 7 8 ================================================================= 0 : 1 1 : 1 1 2 : 3 4 1 3 : 15 21 7 1 4 : 105 148 52 10 1 5 : 945 1333 472 96 13 1 6 : 10395 14664 5197 1066 153 16 1 7 : 135135 190633 67567 13873 2009 223 19 1 8 : 2027025 2859496 1013512 208116 30170 3380 306 22 1 etc.
Crossrefs
Programs
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Mathematica
Flatten[Table[If[k==0,(2n)!/(2^n n!),Sum[Binomial[i-1,k-1]2^i i!/(2i)!,{i,k,n}](2n)!/(2^n n!)],{n,0,8},{k,0,n}]] (* Stefano Spezia, Jan 06 2022 *)
Formula
T(n,n) = 1.
T(n,k) = binomial(n-1,k-1) + (2*n - 1) * T(n-1,k) for 0 < k < n.
Conjecture: M(n,k) = (-1)^(n-k) * T(n,k) is matrix inverse of A350512.