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A245732
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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.
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22
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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Alois P. Heinz, Rows n = 0..140, flattened
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Jul 30 2014
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STATUS
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approved
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