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%I #10 Oct 13 2017 06:07:37
%S 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 28,69,155,197,129,29,1,9,36,106,301,533,571,289,47,1,10,45,154,532,
%U 1223,1884,1640,650,76,1,11,55,215,876,2494,5103,6604,4726,1460,123,1
%N 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 The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.
%C 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Graph Loop.html">Graph Loop</a>.
%e Array begins:
%e . 1 2 3 4 5 6 7 8 9 10
%e . 1 3 6 10 15 21 28 36 45 55
%e . 1 4 11 23 42 69 106 154 215 290
%e . 1 7 26 70 155 301 532 876 1365 2035
%e . 1 11 57 197 533 1223 2494 4654 8105 13355
%e . 1 18 129 571 1884 5103 11998 25362 49347 89848
%e . 1 29 289 1640 6604 21122 57271 137155 298184 599954
%e . 1 47 650 4726 23219 87677 274132 743724 1806597 4016683
%e . 1 76 1460 13604 81555 363606 1310974 4029310 10936124 26868719
%e . 1 123 3281 39175 286555 1508401 6271378 21836366 66220705 179784715
%t (* Run this first: *)
%t << Omega.m;
%t (* Then run the following in a different cell: *)
%t 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 (* Array: *)
%t Grid[Table[a293311[n, k], {n, nmax}, {k, 0, nmax - 1}]]
%t (* Array antidiagonals flattened (gives this sequence): *)
%t Flatten[Table[a293311[n, k - n], {k, 11}, {n, k}]]
%Y Cf. A293311.
%Y Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (rows 1..10).
%K nonn,tabl
%O 1,2
%A _L. Edson Jeffery_, Oct 06 2017
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