A054626 -id:A054626 - cp's OEIS frontend

A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.

7, 28, 28, 119, 637, 119, 616, 19684, 19684, 616, 3367, 721525, 4484039, 721525, 3367, 19684, 28249228, 1153450872, 1153450872, 28249228, 19684, 117655, 1153470437, 316504102999, 2077059243301, 316504102999, 1153470437, 117655, 720916

Offset: 1

R. H. Hardin, Jan 11 2011

			Table starts
       7         28          119           616         3367          19684
      28        637        19684        721525     28249228     1153470437
     119      19684      4484039    1153450872 316504102999 90467424400444
     616     721525   1153450872 2077059243301
    3367   28249228 316504102999
   19684 1153470437
  117655

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 31 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Columns 1-3 are A054626, A184329, A184330.

Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*7^(n*(k/LCM[c, d])), {d, Divisors[k]}], {c, Divisors[n]}]; Table[T[n - k + 1, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 7^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 7^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A054614 a(n) = Sum_{d|n} phi(d)*7^(n/d).

Original entry on oeis.org

0, 7, 56, 357, 2464, 16835, 118104, 823585, 5767328, 40354335, 282492280, 1977326813, 13841410464, 96889010491, 678223896728, 4747561544985, 33232936339456, 232630513987319, 1628413638500376, 11398895185373269

Offset: 0

N. J. A. Sloane, Apr 16 2000

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=7 of A185651.

Cf. A054626.

a(n) = sumdiv(n, d, eulerphi(d)*7^(n/d)); \\ Michel Marcus, Jul 11 2021

a(n) = Sum_{k=1..n} 7^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A121775 T(n, k) = Sum_{d|n} phi(n/d)*binomial(d,k) for n>0, T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 5, 3, 1, 4, 8, 7, 4, 1, 5, 9, 10, 10, 5, 1, 6, 15, 20, 21, 15, 6, 1, 7, 13, 21, 35, 35, 21, 7, 1, 8, 20, 36, 60, 71, 56, 28, 8, 1, 9, 21, 42, 86, 126, 126, 84, 36, 9, 1, 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1, 11, 21, 55, 165, 330, 462, 462, 330, 165, 55

Offset: 0

Paul D. Hanna, Aug 23 2006

For n>0, (1/n)*Sum_{k=0..n} T(n,k)*(c-1)^k is the number of n-bead necklaces with c colors. See the cross references.

			Triangle begins:
[ 0]  1;
[ 1]  1,  1;
[ 2]  2,  3,  1;
[ 3]  3,  5,  3,   1;
[ 4]  4,  8,  7,   4,   1;
[ 5]  5,  9, 10,  10,   5,   1;
[ 6]  6, 15, 20,  21,  15,   6,   1;
[ 7]  7, 13, 21,  35,  35,  21,   7,   1;
[ 8]  8, 20, 36,  60,  71,  56,  28,   8,  1;
[ 9]  9, 21, 42,  86, 126, 126,  84,  36,  9,  1;
[10] 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1;

Cf. A053635 (row sums), A121776 (antidiagonal sums), A054630, A327029.

Cf. A000031 (c=2), A001867 (c=3), A001868 (c=4), A001869 (c=5), A054625 (c=6), A054626 (c=7), A054627 (c=8), A054628 (c=9), A054629 (c=10).

PARI
```
T(n,k)=if(n
				
```

# uses[DivisorTriangle from A327029]
DivisorTriangle(euler_phi, binomial, 13) # Peter Luschny, Aug 24 2019

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A184331 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..6 arrays.

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A054614 a(n) = Sum_{d|n} phi(d)*7^(n/d).

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A121775 T(n, k) = Sum_{d|n} phi(n/d)*binomial(d,k) for n>0, T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

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