A350266 Triangle read by rows. T(n, k) = binomial(n, k) * n! / (n - k + 1)! if k >= 1, if k = 0 then T(n, k) = k^n. T(n, k) for 0 <= k <= n.
1, 0, 1, 0, 2, 2, 0, 3, 9, 6, 0, 4, 24, 48, 24, 0, 5, 50, 200, 300, 120, 0, 6, 90, 600, 1800, 2160, 720, 0, 7, 147, 1470, 7350, 17640, 17640, 5040, 0, 8, 224, 3136, 23520, 94080, 188160, 161280, 40320, 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880
Offset: 0
Examples
Table starts: [0] 1; [1] 0, 1; [2] 0, 2, 2; [3] 0, 3, 9, 6; [4] 0, 4, 24, 48, 24; [5] 0, 5, 50, 200, 300, 120; [6] 0, 6, 90, 600, 1800, 2160, 720; [7] 0, 7, 147, 1470, 7350, 17640, 17640, 5040; [8] 0, 8, 224, 3136, 23520, 94080, 188160, 161280, 40320; [9] 0, 9, 324, 6048, 63504, 381024, 1270080, 2177280, 1632960, 362880;
Crossrefs
Programs
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Maple
T := (n, k) -> ifelse(k = 0, k^n, binomial(n, k)^2 * k! / (n - k + 1)): seq(seq(T(n, k), k = 0..n), n = 0..9);
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Mathematica
T[n_, 0] := Boole[n == 0]; T[n_, k_] := Binomial[n, k]^2 * k!/(n - k + 1); Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 09 2022 *)
Formula
T(n, k) = binomial(n, k)^2 * k! / (n - k + 1) if k >= 1.