A293312 - OEIS

A293312

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.

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, 57, 70, 42, 21, 4, 8, 1, 29, 129, 197, 155, 69, 28, 4, 9, 1, 47, 289, 571, 533, 301, 106, 36, 5, 10, 1, 76, 650, 1640, 1884, 1223, 532, 154, 45, 5, 11

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OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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.

EXAMPLE

Array begins:

. 1 1 1 1 1 1 1 1 1 1 1

. 2 1 3 4 7 11 18 29 47 76 123

. 3 2 6 11 26 57 129 289 650 1460 3281

. 4 2 10 23 70 197 571 1640 4726 13604 39175

. 5 3 15 42 155 533 1884 6604 23219 81555 286555

. 6 3 21 69 301 1223 5103 21122 87677 363606 1508401

. 7 4 28 106 532 2494 11998 57271 274132 1310974 6271378

. 8 4 36 154 876 4654 25362 137155 743724 4029310 21836366

. 9 5 45 215 1365 8105 49347 298184 1806597 10936124 66220705

. 10 5 55 290 2035 13355 89848 599954 4016683 26868719 179784715

. 11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731

. ...

MATHEMATICA

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}]];

(* Array: *)

Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]

(* Array antidiagonals flattened (gives this sequence): *)

Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]

CROSSREFS

Cf. A293311.

Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).

Sequence in context: A326409 A326408 A214717 * A136405 A210871 A308399

Adjacent sequences: A293309 A293310 A293311 * A293313 A293314 A293315

KEYWORD

nonn,tabl

AUTHOR

L. Edson Jeffery, Oct 10 2017

STATUS

approved