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A286096 Triangle read by rows giving numerators of the Fourier expansion of cos^n(x). 0
1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 4, 0, 1, 0, 10, 0, 5, 0, 1, 10, 0, 15, 0, 6, 0, 1, 0, 35, 0, 21, 0, 7, 0, 1, 35, 0, 56, 0, 28, 0, 8, 0, 1, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 126, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1, 0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1, 462, 0, 792, 0, 495, 0, 220, 0, 66, 0, 12, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
Doubling the initial term of each line and dropping the 0's transforms this triangle to the right half of Pascal's triangle (A007318).
Row sums are A011782. - Omar E. Pol, May 02 2017
LINKS
FORMULA
cos^n(x) = (1/2^(n-1)) * Sum_{k=0..n} T(n,k) * cos(k*x).
T(n,k) = T(n-1,k-1) + T(n-1,k+1) if k != 1, T(n,1) = 2*T(n-1,0) + T(n-1,2), T(n,k) = 0 if k < 0 or k > n.
EXAMPLE
Triangle begins:
1;
0, 1;
1, 0, 1;
0, 3, 0, 1;
3, 0, 4, 0, 1;
0, 10, 0, 5, 0, 1;
10, 0, 15, 0, 6, 0, 1;
0, 35, 0, 21, 0, 7, 0, 1;
35, 0, 56, 0, 28, 0, 8, 0, 1;
0, 126, 0, 84, 0, 36, 0, 9, 0, 1;
126, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1;
0, 462, 0, 330, 0, 165, 0, 55, 0, 11, 0, 1;
462, 0, 792, 0, 495, 0, 220, 0, 66, 0, 12, 0, 1;
...
MATHEMATICA
row[n_] := If[n==0, {1}, 2^(n-1)*TrigReduce[Cos[x]^n] /. Cos[Times[k_., x]] -> x^k // CoefficientList[#, x]&]; Table[row[n], {n, 0, 12}] // Flatten
(* Second program: *)
T[n_, n_] = 1; T[n_, k_] /; k == n-1 || k>n = 0; T[n_, 1] := 2 T[n-1, 0] + T[n-1, 2]; T[n_, 0] := T[n-1, 1]; T[n_, k_] /; 1 < k <= n := T[n, k] = T[n-1, k-1] + T[n-1, k+1]; T[_, _] = 0; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 02 2017 *)
CROSSREFS
Cf. A007318, A100257 (same sequence with rows reversed).
Sequence in context: A101000 A035653 A126595 * A247622 A256037 A179898
KEYWORD
nonn,tabl
AUTHOR
Landry Salle, May 02 2017
STATUS
approved

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)