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

Author

N. J. A. Sloane, Aug 07 2014

Keywords

• nonn
• tabl

Comments

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

Examples

			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
  ...
		

Links

• 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).

Crossrefs

Cf. A245558.

Programs

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 *)