cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A336452 Triangle read by rows, the Riordan square of the number of achiral m-color cyclic compositions of n (A308747). T(n,k) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 3, 6, 15, 9, 13, 49, 63, 27, 23, 137, 276, 243, 81, 44, 338, 969, 1323, 891, 243, 73, 782, 2950, 5589, 5778, 3159, 729, 131, 1695, 8161, 20097, 28485, 23733, 10935, 2187, 210, 3545, 20966, 64557, 117801, 133569, 93312, 37179, 6561
Offset: 1

Views

Author

Peter Luschny, Aug 09 2020

Keywords

Comments

See the link and A321620 for more information about the Riordan square transform.

Examples

			[1] [   1]
[2] [   3,    3]
[3] [   6,   15,     9]
[4] [  13,   49,    63,    27]
[5] [  23,  137,   276,   243,     81]
[6] [  44,  338,   969,  1323,    891,    243]
[7] [  73,  782,  2950,  5589,   5778,   3159,   729]
[8] [ 131, 1695,  8161, 20097,  28485,  23733, 10935,  2187]
[9] [ 210, 3545, 20966, 64557, 117801, 133569, 93312, 37179, 6561]

Links

Crossrefs

Cf. A308747, A321620.

Programs

  • SageMath
    # using riordan_square_array from A321620, A308747.
riordan_square_array([1, 3, 6, 13, 23, 44, 73, 131, 210])

A032198 "CIK" (necklace, indistinct, unlabeled) transform of 1,2,3,4,...

Original entry on oeis.org

1, 3, 6, 13, 25, 58, 121, 283, 646, 1527, 3601, 8678, 20881, 50823, 124054, 304573, 750121, 1855098, 4600201, 11442085, 28527446, 71292603, 178526881, 447919418, 1125750145, 2833906683, 7144450566, 18036423973
Offset: 1

Views

Author

Christian G. Bower

Keywords

Examples

			From _Petros Hadjicostas_, Jan 07 2018: (Start)
We give some examples to illustrate the theory of C. G. Bower about transforms given in the weblink above. We assume we have boxes of different sizes and colors that we place on a circle to form a necklace. Two boxes of the same size and same color are considered identical (indistinct and unlabeled). We do, however, change the roles of the sequences (a(n): n>=1) and (b(n): n>=1) that appear in the weblink above. We assume (a(n): n>=1) = CIK((b(n): n>=1)).
Since b(1) = 1, b(2) = 2, b(3) = 3, etc., a box that can hold 1 ball only can be of 1 color only, a box that can hold 2 balls only can be one of 2 colors only, a box that can hold 3 balls can be one of 3 colors, and so on.
To prove that a(3) = 6, we consider three cases. In the first case, we have a single box that can hold 3 balls, and thus we have 3 possibilities for the 3 colors the box can be. In the second case, we have a box that can hold 2 balls and a box that can hold 1 ball. Here, we have 2 x 1 = 2 possibilities. In the third case, we have 3 identical boxes, each of which can hold 1 ball. This gives rise to 1 possibility. Hence, a(3) = 3 + 2 + 1 = 6.
To prove that a(4) = 13, we consider 5 cases: a box with 4 balls (4 possibilities), one box with 3 balls and one box with 1 ball (3 possibilities), two identical boxes each with 2 balls (3 possibilities), one box with 2 balls and two identical boxes each with 1 ball (2 possibilities), and four identical boxes each with 1 ball (1 possibility). Thus, a(4) = 4 + 3 + 3 + 2 + 1 = 13.
(End)

Links

Crossrefs

Cf. A000032, A001906, A004146, A005248, A032170, A032287, A088305, A308723, A308747.
Equals A005594(n) - 1.

Programs

  • Mathematica
    nmax = 30;
f[x_] = Sum[n*x^n, {n, 1, nmax}];
gf = Sum[(EulerPhi[n]/n)*Log[1/(1 - f[x^n])] + O[x]^nmax, {n, 1, nmax}]/x;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 29 2018, after Joerg Arndt *)
  • PARI
    N = 66;  x = 'x + O('x^N);
f(x)=sum(n=1, N, n*x^n );
gf = sum(n=1, N, eulerphi(n)/n*log(1/(1-f(x^n)))  );
v = Vec(gf)
/* Joerg Arndt, Jan 21 2013 */

Formula

a(n) = 1/n*Sum_{d divides n} phi(n/d)*A004146(d). - Vladeta Jovovic, Feb 15 2003
From Petros Hadjicostas, Jan 07 2018: (Start)
a(n) = -2 + (1/n)*Sum_{d|n} phi(n/d)*A005248(d) = -2 + (1/n)*Sum_{d|n} phi(n/d)*L(2*d), where L(n) = A000032(n) is the usual Lucas sequence.
G.f.: -Sum_{n >= 1} (phi(n)/n)*log(1 - B(x^n)), where B(x) = x + 2*x^2 + 3*x^3 + 4*x^4 + ... = x/(1-x)^2.
G.f.: -2*x/(1 - x) - Sum_{n>=1} (phi(n)/n)*log(1 - 3*x^n + x^(2*n)).
(End)
From Petros Hadjicostas, Jun 19 2019: (Start)
According to Gibson et al. (2018), a(n) is the number of m-color cyclic compositions of n where each part of size m has m possible colors. This is nothing else than the CIK transform of the sequence 1, 2, 3, 4, ...
Using the theory of Flajolet and Soria (1991), Gibson et al. (2018, Eq. (1.1)) proved that the g.f. of a(n) is Sum_{s >= 1} (phi(s)/s) * log((1 - x^s)^2/(1 - 3*x^s + x^(2*s))), which is exactly the same g.f. as the ones above.
Gibson et al. (2018, p. 3210) also proved that a(n) ~ (2/(3-sqrt(5)))^n/n for large n. See also Chapter 3 in Gibson (2017).
(End)

A032287 "DIK" (bracelet, indistinct, unlabeled) transform of 1,2,3,4,...

Original entry on oeis.org

1, 3, 6, 13, 24, 51, 97, 207, 428, 946, 2088, 4831, 11209, 26717, 64058, 155725, 380400, 936575, 2314105, 5744700, 14300416, 35708268, 89359536, 224121973, 563126689, 1417378191, 3572884062, 9019324297, 22797540648
Offset: 1

Views

Author

Christian G. Bower

Keywords

Comments

From Petros Hadjicostas, Jun 21 2019: (Start)
Under Bower's transforms, the input sequence c = (c(m): m >= 1) describes how each part of size m in a composition is colored. In a composition (ordered partition) of n >= 1, a part of size m is assumed to be colored with one of c(m) colors.
Under the DIK transform, we are dealing with "dihedral compositions" of n >= 1. These are equivalence classes of ordered partitions of n such that two such ordered partitions are equivalent if one can be obtained from the other by rotation or reflection.
If the input sequence is c = (c(m): m >= 1), denote the output sequence under the DIK transform by b = (b(n): n >= 1); i.e., b(n) = (DIK c)(n) for n >= 1. If C(x) = Sum_{m >= 1} c(m)*x^m is the g.f. of the input sequence c, then the g.f. of b = DIK c is Sum_{n >= 1} b(n)*x^n = -(1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - C(x^d)) + (1 + C(x))^2/(4 * (1 - C(x^2))) - (1/4).
For the current sequence (a(n): n >= 1), the input sequence is c(m) = m for all m >= 1. That is, we are dealing with the so-called "m-color dihedral compositions". Here, a(n) is the number of dihedral compositions of n where each part of size m may be colored with one of m colors. For the linear and cyclic versions of such m-color compositions, see Agarwal (2000), Gibson (2017), and Gibson et al. (2018).
Since C(x) = x/(1 - x)^2, we have Sum_{n >= 1} a(n) * x^n = (1/2) * Sum_{d >= 1} (phi(d)/d) * log((1 - x^d)^2 / (1 - 3*x^d + x^(2*d))) + (1/2) * x * (1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2 * (1 + x - x^2) * (1 - x - x^2)), which is the g.f. given by Andrew Howroyd in the PARI program below.
Note that -Sum_{d >= 1} (phi(d)/d) * log (1 - C(x^d)) = Sum_{d >= 1} (phi(d)/d) * log((1 - x^d)^2 / (1 - 3*x^d + x^(2*d))) is the g.f. of the "m-color cyclic compositions" that appear in Gibson (2017) and Gibson et al. (2018). See sequence A032198, which is the CIK transform of sequence (c(m): m >= 1) = (m: m >= 1).
(End)

Links

Crossrefs

Cf. A000032, A000045, A001906, A005594, A032198, A088305, A308747.

Programs

  • Maple
    DIK := proc(L::list)
    local  x,cidx,ncyc,d,gd,g,g2,n ;
    n := nops(L) ;
    g := add(op(i,L)*x^i,i=1..n) ;
    # wrap into the cycle index of the cyclic group C_n
    cidx := 0 ;
    for ncyc from 1 to n do
        for d in numtheory[divisors](ncyc) do
            gd := subs(x=x^d,g) ;
            cidx := cidx+1/ncyc*numtheory[phi](d)*gd^(ncyc/d) ;
        end do:
    end do:
    # cycle index is half of th eone for the cyclic group plus two
    # different branches or D_n with even or odd n
    cidx := cidx/2 ;
    g2 := subs(x=x^2,g) ;
    for ncyc from 1 to n do
        if type(ncyc,'odd') then
            cidx := cidx+ g*g2^((ncyc-1)/2)/2 ;
        else
            cidx := cidx+ (g^2*g2^((ncyc-2)/2) + g2^(ncyc/2))/4 ;
        end if;
    end do:
    taylor(cidx,x=0,nops(L)) ;
    gfun[seriestolist](%) ;
end proc:
A032287_list := proc(n)
        local ele ;
        ele := [seq(i,i=1..40)] ;
        DIK(ele) ;
end proc:
A032287_list(50) ; # R. J. Mathar, Feb 14 2025
  • Mathematica
    seq[n_] := x(1 + x - 2 x^2 + x^3 + x^4)/((1 - x)^2 (1 - x - x^2)(1 + x - x^2)) + Sum[EulerPhi[d]/d Log[(1 - x^d)^2/(1 - 3 x^d + x^(2d)) + O[x]^(n+1)], {d, 1, n}] // CoefficientList[#, x]& // Rest // #/2&;
seq[30] (* Jean-François Alcover, Sep 17 2019, from PARI *)
  • PARI
    seq(n)={Vec(x*(1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2*(1 - x - x^2)*(1 + x - x^2)) + sum(d=1, n, eulerphi(d)/d*log((1 - x^d)^2/(1 - 3*x^d + x^(2*d)) + O(x*x^n))))/2} \\ Andrew Howroyd, Jun 20 2018

Formula

From Petros Hadjicostas, Jun 21 2019: (Start)
a(n) = ( F(n+4) + (-1)^n * F(n-4) - 2 * (n + 1) + (1/n) * Sum_{d|n} phi(n/d) * L(2*d) )/2 for n >= 4, where F(n) = A000045(n) and L(n) = A000032(n) are the usual n-th Fibonacci and n-th Lucas numbers, respectively.
a(n) = (A032198(n) + A308747(n))/2 for n >= 1.
G.f.: (1/2) * Sum_{d >= 1} (phi(d)/d) * log((1 - x^d)^2 / (1 - 3*x^d + x^(2*d))) + (1/2) * x * (1 + x - 2*x^2 + x^3 + x^4)/((1 - x)^2 * (1 + x - x^2) * (1 - x - x^2)).
(End)
Showing 1-3 of 3 results.

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