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%I #34 May 09 2018 10:29:08
%S 1,1,1,4,3,1,27,13,1,1,256,75,7,1,1,3125,541,21,1,1,1,46656,4683,141,
%T 21,1,1,1,823543,47293,743,71,1,1,1,1,16777216,545835,5699,183,71,1,1,
%U 1,1,387420489,7087261,42241,2101,253,1,1,1,1,1
%N 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.
%C T(0,0) = 1 by convention.
%C 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
%H Alois P. Heinz, <a href="/A245732/b245732.txt">Rows n = 0..140, flattened</a>
%F 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
%e Triangle T(n,k) begins:
%e 0 : 1;
%e 1 : 1, 1;
%e 2 : 4, 3, 1;
%e 3 : 27, 13, 1, 1;
%e 4 : 256, 75, 7, 1, 1;
%e 5 : 3125, 541, 21, 1, 1, 1;
%e 6 : 46656, 4683, 141, 21, 1, 1, 1;
%e 7 : 823543, 47293, 743, 71, 1, 1, 1, 1;
%e 8 : 16777216, 545835, 5699, 183, 71, 1, 1, 1, 1;
%p b:= proc(n, k) option remember; `if`(n=0, 1,
%p add(b(n-j, k)*binomial(n, j), j=k..n))
%p end:
%p T:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t 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_ *)
%Y Column k=0 gives A000312.
%Y Columns k=1-10 give (for n>0): A000670, A032032, A102233, A232475, A245790, A245791, A245792, A245793, A245794, A245795.
%Y T(2n,n) gives A244174(n) or 1+A007318(2n,n) = 1+A000984(n) for n>0.
%Y Cf. A245733.
%K nonn,tabl
%O 0,4
%A _Alois P. Heinz_, Jul 30 2014
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