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

%I A245559 #16 May 18 2024 09:53:58
%S A245559 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,
%T A245559 1,4,15,40,99,212,429,800,1,5,18,55,143,333,715,1430,2700,1,5,22,70,
%U A245559 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
%N A245559 Triangle read by rows: entries on or below the main diagonal in A245558.
%C A245559 See A245558 for identification of other sequences occurring in this triangle.
%H A245559 A. Elashvili and M. Jibladze, <a href="http://dx.doi.org/10.1016/S0019-3577(98)80021-9">Hermite reciprocity for the regular representations of cyclic groups</a>, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238; MR1691428 (2000c:13006).
%H A245559 A. Elashvili, M. Jibladze, and D. Pataraia, <a href="http://dx.doi.org/10.1023/A:1018727630642">Combinatorics of necklaces and "Hermite reciprocity"</a>, J. Algebraic Combin. 10 (1999), no. 2, 173-188; MR1719140 (2000j:05009). See p. 174.
%H A245559 J. E. Iglesias, <a href="https://doi.org/10.1524/zkri.2006.221.4.237">Enumeration of closest-packings by the space group: a simple approach</a>, Z. Krist. 221 (2006) 237-245, eq. (5).
%e A245559 Triangle begins:
%e A245559   1,
%e A245559   1, 1,
%e A245559   1, 2, 3,
%e A245559   1, 2, 5, 8,
%e A245559   1, 3, 7, 14, 25,
%e A245559   1, 3, 9, 20, 42, 75,
%e A245559   1, 4, 12, 30, 66, 132, 245,
%e A245559   1, 4, 15, 40, 99, 212, 429, 800,
%e A245559   1, 5, 18, 55, 143, 333, 715, 1430, 2700,
%e A245559   1, 5, 22, 70, 200, 497, 1144, 2424, 4862, 9225,
%e A245559   1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32065,
%e A245559   1, 6, 30, 112, 364, 1026, 2652, 6288, 13995, 29372, 58786, 112632
%e A245559   ...
%p A245559 A245559 := proc(p,q)
%p A245559     local d;
%p A245559     a := 0 ;
%p A245559     for d from 1 to max(p,q) do
%p A245559         if modp(p,d)=0 and modp(q,d)=0 then
%p A245559             a := a+numtheory[mobius](d)*(binomial((p+q)/d,p/d)) ;
%p A245559         end if ;
%p A245559     end do:
%p A245559     a/(p+q) ;
%p A245559 end proc:
%p A245559 seq(seq( A245559(p,q),q=1..p),p=1..12) ; # _R. J. Mathar_, Apr 15 2024
%t A245559 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)];
%t A245559 Table[Table[A245559[p, q], {q, 1, p}], {p, 1, 12}] // Flatten (* _Jean-François Alcover_, May 17 2024, after _R. J. Mathar_ *)
%Y A245559 Cf. A245558.
%K A245559 nonn,tabl
%O A245559 1,5
%A A245559 _N. J. A. Sloane_, Aug 07 2014