A245687 Number T(n,k) of endofunctions on [n] such that the minimal cardinality of the nonempty preimages equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 2, 2, 0, 24, 0, 3, 0, 216, 36, 0, 4, 0, 2920, 200, 0, 0, 5, 0, 44100, 2250, 300, 0, 0, 6, 0, 799134, 22932, 1470, 0, 0, 0, 7, 0, 16429504, 342608, 3136, 1960, 0, 0, 0, 8, 0, 382625856, 4638384, 147168, 9072, 0, 0, 0, 0, 9, 0, 9918836100, 79610850, 1522800, 18900, 11340, 0, 0, 0, 0, 10
Offset: 0
Examples
Triangle T(n,k) begins: 1; 0, 1; 0, 2, 2; 0, 24, 0, 3; 0, 216, 36, 0, 4; 0, 2920, 200, 0, 0, 5; 0, 44100, 2250, 300, 0, 0, 6; 0, 799134, 22932, 1470, 0, 0, 0, 7; 0, 16429504, 342608, 3136, 1960, 0, 0, 0, 8; ...
Links
- Alois P. Heinz, Rows n = 0..100, flattened
Crossrefs
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k) +add(b(n-j, i-1, k)/j!, j=k..n))) end: T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), `if`(k=n, n, `if`(k>=(n+1)/2, 0, n!*(b(n$2, k)-b(n$2, k+1))))): seq(seq(T(n, k), k=0..n), n=0..12); -
Mathematica
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-j, i-1, k]/j!, {j, k, n}]]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == n, n, If[k >= (n+1)/2, 0, n!*(b[n, n, k] - b[n, n, k+1])]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *)
Comments