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A225723 Triangular array read by rows: T(n,k) is the number of size k components in the digraph representation of all functions f:{1,2,...,n}->{1,2,...,n}; n>=1, 1<=k<=n. 1
1, 2, 3, 12, 9, 17, 108, 72, 68, 142, 1280, 810, 680, 710, 1569, 18750, 11520, 9180, 8520, 9414, 21576, 326592, 196875, 152320, 134190, 131796, 151032, 355081, 6588344, 3919104, 2975000, 2544640, 2372328, 2416512, 2840648, 6805296 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
T(n,1) = n*(n-1)^(n-1) = A055897(n).
Row sums = A190314.
T(n,n) = A001865(n).
Sum_{k=1..n} T(n,k)*k = n^(n+1).
LINKS
FORMULA
E.g.f.: log(1/(1 - A(x*y)))/(1 - A(x)) where A(x) is the e.g.f. for A000169.
T(n,k) = C(n,k)*A001865(k)*A000312(n-k). - Alois P. Heinz, May 13 2013
EXAMPLE
Triangle T(n,k) begins:
1;
2, 3;
12, 9, 17;
108, 72, 68, 142;
1280, 810, 680, 710, 1569;
18750, 11520, 9180, 8520, 9414, 21576;
326592, 196875, 152320, 134190, 131796, 151032, 355081;
...
MAPLE
b:= n-> n!*add(n^(n-k-1)/(n-k)!, k=1..n):
T:= (n, k)-> binomial(n, k)*b(k)*(n-k)^(n-k):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, May 13 2013
MATHEMATICA
nn = 8; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; txy =
Sum[n^(n - 1) (x y)^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
Drop[Range[0, nn]! CoefficientList[
Series[Log[1/(1 - txy)]/(1 - tx), {x, 0, nn}], {x, y}],
1]] // Grid
CROSSREFS
Cf. A225213.
Sequence in context: A320810 A104038 A112979 * A092972 A334914 A261576
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, May 13 2013
STATUS
approved

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Last modified February 23 14:24 EST 2024. Contains 370283 sequences. (Running on oeis4.)