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A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.

%I A293311 #11 Feb 16 2025 08:33:51
%S A293311 1,2,1,3,3,1,4,6,4,1,5,10,11,7,1,6,15,23,26,11,1,7,21,42,70,57,18,1,8,
%T A293311 28,69,155,197,129,29,1,9,36,106,301,533,571,289,47,1,10,45,154,532,
%U A293311 1223,1884,1640,650,76,1,11,55,215,876,2494,5103,6604,4726,1460,123,1
%N A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.
%C A293311 The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.
%C A293311 The generating function for row n of this array was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews, et al., and is needed for the Mathematica program below.
%H A293311 G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis III. The Omega package</a>.
%H A293311 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>.
%H A293311 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Graph Loop.html">Graph Loop</a>.
%e A293311 Array begins:
%e A293311 .  1   2    3     4      5       6       7        8        9        10
%e A293311 .  1   3    6    10     15      21      28       36       45        55
%e A293311 .  1   4   11    23     42      69     106      154      215       290
%e A293311 .  1   7   26    70    155     301     532      876     1365      2035
%e A293311 .  1  11   57   197    533    1223    2494     4654     8105     13355
%e A293311 .  1  18  129   571   1884    5103   11998    25362    49347     89848
%e A293311 .  1  29  289  1640   6604   21122   57271   137155   298184    599954
%e A293311 .  1  47  650  4726  23219   87677  274132   743724  1806597   4016683
%e A293311 .  1  76 1460 13604  81555  363606 1310974  4029310 10936124  26868719
%e A293311 .  1 123 3281 39175 286555 1508401 6271378 21836366 66220705 179784715
%t A293311 (* Run this first: *)
%t A293311 << Omega.m;
%t A293311 (* Then run the following in a different cell: *)
%t A293311 nmax = 11; Do[cond = {}; Do[If[n == 1, AppendTo[cond, Subscript[a, 1] + Subscript[a, 2] == Subscript[a, 3]]; Break[], AppendTo[cond, If[j == n, Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 1] == Subscript[a, 2*n + 1], Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 2*j + 1] == Subscript[a, 2*n + 1]]]], {j, n}]; f = OEqSum[Product[Subscript[x, i]^Subscript[a, i], {i, 2*n + 1}], cond, \[Lambda]][[1]] /. {Subscript[x, 2*n + 1] -> z} /. {Subscript[x, _] -> 1}; Do[f = OEqR[f, Subscript[\[Lambda], k]], {k, Length[cond]}]; Do[a293311[n, k] = Coefficient[Series[f, {z, 0, nmax - 1}], z, k], {k, 0, nmax - 1}], {n, nmax}];
%t A293311 (* Array: *)
%t A293311 Grid[Table[a293311[n, k], {n, nmax}, {k, 0, nmax - 1}]]
%t A293311 (* Array antidiagonals flattened (gives this sequence): *)
%t A293311 Flatten[Table[a293311[n, k - n], {k, 11}, {n, k}]]
%Y A293311 Cf. A293311.
%Y A293311 Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (rows 1..10).
%K A293311 nonn,tabl
%O A293311 1,2
%A A293311 _L. Edson Jeffery_, Oct 06 2017