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A245558 Square array read by antidiagonals: T(n,k) = number of n-tuples of nonnegative integers (u_0,...,u_{n-1}) satisfying Sum_{j=0..n-1} j*u_j == 1 mod n and Sum_{j=0..n-1} u_j = m. 5
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 3, 7, 8, 7, 3, 1, 1, 4, 9, 14, 14, 9, 4, 1, 1, 4, 12, 20, 25, 20, 12, 4, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 5, 18, 40, 66, 75, 66, 40, 18, 5, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

The array is symmetric; for the entries on or below the diagonal see A245559.

If the congruence in the definition is changed from Sum_{j=0..n-1} j*u_j == 1 mod n  to Sum_{j=0..n-1} j*u_j == 0 mod n we get the array shown in A241926, A047996, and A037306.

REFERENCES

Elashvili, A.; Jibladze, M.; Hermite reciprocity for the regular representations of cyclic groups. Indag. Math. (N.S.) 9 (1998), no. 2, 233--238. MR1691428 (2000c:13006).

Elashvili, A.; Jibladze, M.; Pataraia, D. Combinatorics of necklaces and "Hermite reciprocity''. J. Algebraic Combin. 10 (1999), no. 2, 173--188. MR1719140 (2000j:05009). See p. 174.

I. M. Gessel, C. Reuenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory A 64 (1993) 189-215, Theorem 9.4.

LINKS

Table of n, a(n) for n=1..77.

EXAMPLE

Square array begins:

1,1,1,1,1,1,1,1,1,1,...

1,1,2,2,3,3,4,4,5,5,...

1,2,3,5,7,9,12,15,18,22,...

1,2,5,8,14,20,30,40,55,70,...

1,3,7,14,25,42,66,99,143,200,...

1,3,9,20,42,75,132,212,333,497,...

1,4,12,30,66,132,245,429,715,1144,...

1,4,15,40,99,212,429,800,1430,2424,...

1,5,18,55,143,333,715,1430,2700,4862,...

1,5,22,70,200,497,1144,2424,4862,9225,...

...

Reading by antidiagonals, we get:

1,

1,1,

1,1,1,

1,2,2,1,

1,2,3,2,1,

1,3,5,5,3,1,

1,3,7,8,7,3,1,

1,4,9,14,14,9,4,1,

1,4,12,20,25,20,12,4,1,

1,5,15,30,42,42,30,15,5,1,

1,5,18,40,66,75,66,40,18,5,1,

1,6,22,55,99,132,132,99,55,22,1,

...

MAPLE

# To produce the first 10 rows and columns (as on page 174 of the Elashvili et al. 1999 reference):

with(numtheory):

cnk:=(n, k) -> add(mobius(n/d)*d, d in divisors(gcd(n, k)));

anmk:=(n, m, k)->(1/(n+m))*add( cnk(d, k)*binomial((n+m)/d, n/d), d in divisors(gcd(n, m))); # anmk(n, m, k) is the value of a_k(n, m) as in Theorem 1, Equation (4), of the Elashvili et al. 1999 reference.

r2:=(n, k)->[seq(anmk(n, m, k), m=1..10)];

for n from 1 to 10 do lprint(r2(n, 1)); od:

CROSSREFS

This array is very similar to but different from A011847.

Cf. A051168, A241926, A047996, A037306, A245559.

Rows include A001840, A006918, A051170, A011796, A011797, A031164. Main diagonal is A022553.

Sequence in context: A176298 A259575 A169623 * A011847 A091325 A193596

Adjacent sequences:  A245555 A245556 A245557 * A245559 A245560 A245561

KEYWORD

nonn,tabl,changed

AUTHOR

N. J. A. Sloane, Aug 07 2014

STATUS

approved

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Last modified July 23 08:59 EDT 2017. Contains 289686 sequences.