A245559 Triangle read by rows: entries on or below the main diagonal in A245558.

1, 1, 1, 1, 2, 3, 1, 2, 5, 8, 1, 3, 7, 14, 25, 1, 3, 9, 20, 42, 75, 1, 4, 12, 30, 66, 132, 245, 1, 4, 15, 40, 99, 212, 429, 800, 1, 5, 18, 55, 143, 333, 715, 1430, 2700, 1, 5, 22, 70, 200, 497, 1144, 2424, 4862, 9225, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32065, 1, 6, 30, 112, 364, 1026, 2652, 6288, 13995, 29372, 58786, 112632

Offset

1,5

Comments

See A245558 for identification of other sequences occurring in this triangle.

Links

Table of n, a(n) for n=1..78.

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

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

J. E. Iglesias, Enumeration of closest-packings by the space group: a simple approach, Z. Krist. 221 (2006) 237-245, eq. (5).

Example

Triangle begins:
  1,
  1, 1,
  1, 2, 3,
  1, 2, 5, 8,
  1, 3, 7, 14, 25,
  1, 3, 9, 20, 42, 75,
  1, 4, 12, 30, 66, 132, 245,
  1, 4, 15, 40, 99, 212, 429, 800,
  1, 5, 18, 55, 143, 333, 715, 1430, 2700,
  1, 5, 22, 70, 200, 497, 1144, 2424, 4862, 9225,
  1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32065,
  1, 6, 30, 112, 364, 1026, 2652, 6288, 13995, 29372, 58786, 112632
  ...

Maple

A245559 := proc(p, q)
    local d;
    a := 0 ;
    for d from 1 to max(p, q) do
        if modp(p, d)=0 and modp(q, d)=0 then
            a := a+numtheory[mobius](d)*(binomial((p+q)/d, p/d)) ;
        end if ;
    end do:
    a/(p+q) ;
end proc:
seq(seq( A245559(p, q), q=1..p), p=1..12) ; # R. J. Mathar, Apr 15 2024

Mathematica

A245559[p_, q_] := Module[{d, a = 0}, For[d = 1, d <= Max[p, q], d++, If[Mod[p, d] == 0 && Mod[q, d] == 0, a = a + MoebiusMu[d]*Binomial[ (p+q)/d, p/d]]]; a/(p+q)];
Table[Table[A245559[p, q], {q, 1, p}], {p, 1, 12}] // Flatten (* Jean-François Alcover, May 17 2024, after R. J. Mathar *)

Crossrefs

Cf. A245558.

Sequence in context: A306732 A109202 A362599 * A117488 A307668 A308173

Adjacent sequences: A245556 A245557 A245558 * A245560 A245561 A245562

Keyword

nonn,tabl,changed

Author

N. J. A. Sloane, Aug 07 2014

Status

approved