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 A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 22
 1, 1, 1, 4, 3, 1, 27, 13, 1, 1, 256, 75, 7, 1, 1, 3125, 541, 21, 1, 1, 1, 46656, 4683, 141, 21, 1, 1, 1, 823543, 47293, 743, 71, 1, 1, 1, 1, 16777216, 545835, 5699, 183, 71, 1, 1, 1, 1, 387420489, 7087261, 42241, 2101, 253, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS T(0,0) = 1 by convention. In general, column k > 1 is asymptotic to n! / ((1+r^(k-1)/(k-1)!) * r^(n+1)), where r is the root of the equation 2 - exp(r) + Sum_{j=1..k-1} r^j/j! = 0. - Vaclav Kotesovec, Aug 02 2014 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA E.g.f. (for column k > 0): 1/(2 -exp(x) +Sum_{j=1..k-1} x^j/j!) -1. - Vaclav Kotesovec, Aug 02 2014 EXAMPLE Triangle T(n,k) begins: 0 : 1; 1 : 1, 1; 2 : 4, 3, 1; 3 : 27, 13, 1, 1; 4 : 256, 75, 7, 1, 1; 5 : 3125, 541, 21, 1, 1, 1; 6 : 46656, 4683, 141, 21, 1, 1, 1; 7 : 823543, 47293, 743, 71, 1, 1, 1, 1; 8 : 16777216, 545835, 5699, 183, 71, 1, 1, 1, 1; MAPLE b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*binomial(n, j), j=k..n)) end: T:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; T[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 05 2015, after Alois P. Heinz *) CROSSREFS Column k=0 gives A000312. Columns k=1-10 give (for n>0): A000670, A032032, A102233, A232475, A245790, A245791, A245792, A245793, A245794, A245795. T(2n,n) gives A244174(n) or 1+A007318(2n,n) = 1+A000984(n) for n>0. Cf. A245733. Sequence in context: A350528 A208057 A298673 * A039621 A142158 A203412 Adjacent sequences: A245729 A245730 A245731 * A245733 A245734 A245735 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 30 2014 STATUS approved

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Last modified March 22 18:15 EDT 2023. Contains 361432 sequences. (Running on oeis4.)