login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A075195 Jablonski table T(n,k) read by antidiagonals: T(n,k) = number of necklaces with n beads of k colors. 25
1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 11, 6, 1, 6, 15, 24, 24, 8, 1, 7, 21, 45, 70, 51, 14, 1, 8, 28, 76, 165, 208, 130, 20, 1, 9, 36, 119, 336, 629, 700, 315, 36, 1, 10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1, 11, 55, 249, 1044, 3367, 7826, 11165, 8230, 2195, 108, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 86 (2.2.23).

Louis Comtet, Analyse combinatoire, Tome 2, p. 104 #17, P.U.F., 1970.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

E. Jablonski, Théorie des permutations et des arrangements complets, Journal de Liouville, 8 (1892), pp. 331-49.

Index entries for sequences related to necklaces

FORMULA

T(n, k) = (1/n)* Sum_{d divides n}*phi(d)*k^(n/d), where phi = Euler totient function A000010. - Philippe Deléham, Oct 08 2003

EXAMPLE

The array T(n,k) for n >= 1, k >= 1 begins:

  1, 2,  3,   4,   5, ...

  1, 3,  6,  10,  15, ...

  1, 4, 11,  24,  45, ...

  1, 6, 24,  70, 165, ...

  1, 8, 51, 208, 629, ...

From Indranil Ghosh, Mar 25 2017: (Start)

Triangle formed when the array is read by antidiagonals:

    1

    2,  1

    3,  3,   1

    4,  6,   4,   1

    5, 10,  11,   6,    1

    6, 15,  24,  24,    8,    1

    7, 21,  45,  70,   51,   14,    1

    8, 28,  76, 165,  208,  130,   20,   1

    9, 36, 119, 336,  629,  700,  315,  36,  1

   10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1

   ...

(End)

MATHEMATICA

t[n_, k_] := (1/n)*Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[t[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Philippe Deléham *)

PROG

(PARI) T(n, k) = (1/n) * sumdiv(n, d, eulerphi(d)*k^(n/d));

for(n=1, 15, for(k=1, n, print1(T(k, n - k + 1), ", "); ); print(); ) \\ Indranil Ghosh, Mar 25 2017

(Python)

from sympy.ntheory import totient, divisors

def T(n, k): return (1/n)*sum([totient(d)*k**(n/d) for d in divisors(n)])

for n in xrange(1, 16):

....print [T(k, n - k + 1) for k in xrange(1, n + 1)] # Indranil Ghosh, Mar 25 2017

CROSSREFS

Columns 1-10: A000012, A000031, A001867, A001868, A001869, A054625-A054629.

Rows 1-10: A000027, A000217, A006527, A006528, A054620, A006565, A054621-A054624.

Main Diagonal: A056665. A054630 and A054631 are the upper and lower triangles.

Cf. A000010.

Sequence in context: A074909 A135278 A034356 * A293311 A126885 A239986

Adjacent sequences:  A075192 A075193 A075194 * A075196 A075197 A075198

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Sep 07 2002

EXTENSIONS

Additional references from Philippe Deléham, Oct 08 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 19 11:04 EST 2018. Contains 299330 sequences. (Running on oeis4.)