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A081720 Triangle T(n,k) read by rows, giving number of bracelets (turn over necklaces) with n beads of k colors (n >= 1, 1 <= k <= n). 19
1, 1, 3, 1, 4, 10, 1, 6, 21, 55, 1, 8, 39, 136, 377, 1, 13, 92, 430, 1505, 4291, 1, 18, 198, 1300, 5895, 20646, 60028, 1, 30, 498, 4435, 25395, 107331, 365260, 1058058, 1, 46, 1219, 15084, 110085, 563786, 2250311, 7472984, 21552969, 1, 78, 3210, 53764, 493131, 3037314 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

From Petros Hadjicostas, Nov 29 2017: (Start)

The formula given below is clear from the programs given in the Maple and Mathematica sections, while the g.f. for column k can be obtained using standard techniques.

If we differentiate the column k g.f. m times, then we can get a formula for row m. (For this sequence, we only need to use this row m formula for 1 <= k <= m, but it is valid even for k>m.) For example, to get the formula for row 8, we have T(n=8,k) = d^8/dx^8 (column k g.f.)/8! evaluated at x=0. Here, "d^8/dx^8" means "8-th derivative w.r.t. x" of the column k g.f. Doing so, we get T(n=8, k) = (k^6 - k^5 + k^4 + 3*k^3 + 2*k^2 - 2*k + 4)*(k + 1)*k/16, which is the formula given for sequence A060560. (Here, we use this formula only for 1 <= k <= 8.)

(End)

LINKS

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

FORMULA

See Maple code.

From Petros Hadjicostas, Nov 29 2017: (Start)

T(n,k) = ((1+k)*k^{n/2}/2 + (1/n)*Sum_{d|n} phi(n/d)*k^d)/2, if n is even, and = (k^{(n+1)/2} + (1/n)*Sum_{d|n} phi(n/d)*k^d)/2, if n is odd.

G.f. for column k: (1/2)*((k*x+k*(k+1)*x^2/2)/(1-k*x^2) - Sum_{n>=1} (phi(n)/n)*log(1-k*x^n)) provided we chop off the Taylor expansion starting at x^k (and ignore all the terms x^n with n<k).

(End)

EXAMPLE

1;                                                (A000027)

1,  3;                                            (A000217)

1,  4,  10;                                       (A000292)

1,  6,  21,   55;                                 (A002817)

1,  8,  39,  136,   377;                          (A060446)

1, 13,  92,  430,  1505,   4291;                  (A027670)

1, 18, 198, 1300,  5895,  20646,  60028;          (A060532)

1, 30, 498, 4435, 25395, 107331, 365260, 1058058; (A060560)

...

For example, when n=k=3, we have the following T(3,3)=10 bracelets of 3 beads using up to 3 colors: 000, 001, 002, 011, 012, 022, 111, 112, 122, and 222. (Note that 012 = 120 = 201 = 210 = 102 = 021.) Petros Hadjicostas, Nov 29 2017

MAPLE

with(numtheory); T := proc(n, a) local d, t1; if n mod 2 = 0 then t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+phi(d)*a^(n/d); fi; od; RETURN((t1+(n/2)*(1+a)*a^(n/2))/(2*n)); else t1 := 0; for d from 1 to n do if n mod d = 0 then t1 := t1+phi(d)*a^(n/d); fi; od; RETURN((t1+n*a^((n+1)/2))/(2*n)); fi; end;

MATHEMATICA

t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Sep 13 2012, after Maple, updated Nov 02 2017 *)

Needs["Combinatorica`"]; Table[Table[NumberOfNecklaces[n, k, Dihedral], {k, 1, n}], {n, 1, 8}]//Grid  (* Geoffrey Critzer, Oct 07 2012, after code by T. D. Noe in A027671 *)

CROSSREFS

Cf. A081721 (diagonal), A081722 (row sums), column sequences k=2..6: A000029, A027671, A032275, A032276, A056341.

Sequence in context: A190179 A025116 A178300 * A137405 A262078 A121922

Adjacent sequences:  A081717 A081718 A081719 * A081721 A081722 A081723

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, based on information supplied by Gary W. Adamson, Apr 05 2003

EXTENSIONS

Name edited by Petros Hadjicostas, Nov 29 2017.

STATUS

approved

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Last modified February 25 18:44 EST 2018. Contains 299655 sequences. (Running on oeis4.)