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A071207
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Triangular array T(n,k) read by rows, giving number of labeled free trees with n vertices and k children of the root that have a label smaller than the label of the root.
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5
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1, 1, 1, 4, 4, 1, 27, 27, 9, 1, 256, 256, 96, 16, 1, 3125, 3125, 1250, 250, 25, 1, 46656, 46656, 19440, 4320, 540, 36, 1, 823543, 823543, 352947, 84035, 12005, 1029, 49, 1, 16777216, 16777216, 7340032, 1835008, 286720, 28672, 1792, 64, 1, 387420489
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| The n-th term of the n-th binomial transform of a sequence {b} is given by {d} where d(n) = sum(k=0,n,T(n,k)*b(k)) and T(n,k)=binomial(n,k)*n^(n-k); such diagonals are related to the hyperbinomial transform (A088956). - Paul D. Hanna (pauldhanna(AT)juno.com), Nov 04 2003
T(n,k) gives the number of divisors of A181555(n) with (n-k) distinct prime factors. See also A001221, A146289, A146290, A181567. [From Matthew Vandermast (ghodges14(AT)comcast.net), Oct 31 2010]
T(n,k) is the number of partial functions on {1,2,...,n} leaving exactly k elements undefined. Row sums = A000169. - Geoffrey Critzer, Jan 08 2012.
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REFERENCES
| C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
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FORMULA
| T(n,k) = binomial(n, k)*n^(n-k).
E.g.f.: (-LambertW(-y)/y)^x/(1+LambertW(-y)). - Vladeta Jovovic (vladeta(AT)eunet.rs)
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EXAMPLE
| 1
1 1
4 4 1
27 27 9 1
256 256 96 16 1
3125 3125 1250 250 25 1
46656 46656 19440 4320 540 36 1
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MAPLE
| T:= (n, k)-> binomial(n, k)*n^(n-k): seq(seq(T(n, k), k=0..n), n=0..10);
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MATHEMATICA
| Prepend[Flatten[ Table[Table[Binomial[n, k] n^(n - k), {k, 0, n}], {n, 1, 8}]], 1] (*Geoffrey Critzer, Jan 08 2012*)
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PROG
| (PARI) T(n, k)=if(k<0|k>n, 0, binomial(n, k)*n^(n-k))
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CROSSREFS
| Cf. A000312.
Cf. A089466, A088956.
Sequence in context: A116866 A126280 A170986 * A136214 A067328 A111845
Adjacent sequences: A071204 A071205 A071206 * A071208 A071209 A071210
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Cedric Chauve (chauve(AT)lacim.uqam.ca), May 16 2002
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