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%I #13 Oct 13 2017 06:07:56
%S 1,1,2,1,1,3,1,3,2,4,1,4,6,2,5,1,7,11,10,3,6,1,11,26,23,15,3,7,1,18,
%T 57,70,42,21,4,8,1,29,129,197,155,69,28,4,9,1,47,289,571,533,301,106,
%U 36,5,10,1,76,650,1640,1884,1223,532,154,45,5,11
%N Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.
%C Conjecture: For all n >= 1, for all k >= 2, A(n, k) = A293311(k, n); i.e., A(n, k) = number of magic labelings of the graph LOOP X C_k with magic sum n - 1.
%F Let S(0, x) = 1, S(1, x) = x, S(k, x) = x*S(k - 1, x) - S(k - 2, x) (the S-polynomials of _Wolfdieter Lang_) and c(n, j) = 2*(-1)^(j - 1)*cos(j*Pi/(2*n + 1)). Then A(n, k) = Sum_{j=1..n} S(n - 1, c(n, j))^(k), n >= 1, k >= 0.
%e Array begins:
%e . 1 1 1 1 1 1 1 1 1 1 1
%e . 2 1 3 4 7 11 18 29 47 76 123
%e . 3 2 6 11 26 57 129 289 650 1460 3281
%e . 4 2 10 23 70 197 571 1640 4726 13604 39175
%e . 5 3 15 42 155 533 1884 6604 23219 81555 286555
%e . 6 3 21 69 301 1223 5103 21122 87677 363606 1508401
%e . 7 4 28 106 532 2494 11998 57271 274132 1310974 6271378
%e . 8 4 36 154 876 4654 25362 137155 743724 4029310 21836366
%e . 9 5 45 215 1365 8105 49347 298184 1806597 10936124 66220705
%e . 10 5 55 290 2035 13355 89848 599954 4016683 26868719 179784715
%e . 11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731
%e . ...
%t s[0, x_] := 1; s[1, x_] := x; s[k_, x_] := x*s[k - 1, x] - s[k - 2, x]; c[n_, j_] := 2 (-1)^(j - 1) Cos[j*Pi/(2 n + 1)]; a[n_, k_] := Round[Sum[s[n - 1, c[n, j]]^(k), {j, n}]];
%t (* Array: *)
%t Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]
%t (* Array antidiagonals flattened (gives this sequence): *)
%t Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]
%Y Cf. A293311.
%Y Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).
%Y Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).
%K nonn,tabl
%O 1,3
%A _L. Edson Jeffery_, Oct 10 2017
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