A273167 Numerators of coefficient triangle for expansion of x^(2*n) in terms of Chebyshev polynomials of the first kind T(2*m, x) (A127674).

1, 1, 1, 3, 1, 1, 5, 15, 3, 1, 35, 7, 7, 1, 1, 63, 105, 15, 45, 5, 1, 231, 99, 495, 55, 33, 3, 1, 429, 3003, 1001, 1001, 91, 91, 7, 1, 6435, 715, 1001, 273, 455, 35, 15, 1, 1, 12155, 21879, 1989, 4641, 1071, 765, 51, 153, 9, 1, 46189, 20995, 62985, 4845, 4845, 969, 4845, 285, 95, 5, 1

Offset

0,4

Comments

The denominator triangle is given in A273168.

The expansion is x^(2*n) = Sum_{m=0..n} R(n, m)*Tnx(2*m, x), n >= 0, with the rational triangle R(n, m) = a(n, m)/A273168(n, m).

Compare this with A127673.

This is equivalent to the expansion cos(x)^(2n) = Sum_{m=0..n} R(n, m)*cos(2*m*x), n >= 0. Compare this with the even numbered rows of A273496.

See A244420/A244421 for the expansion of x^(2*n+1) in terms of odd-indexed Chebyshev polynomials of the first kind.

The signed rational triangle S(n, m) = R(n, m) * (-1)^m appears in the expansion sin(x)^(2n) = Sum_{m=0..n} S(n, m) * cos(2*m*x), n >= 0. This is equivalent to the identity (1-x^2)^n = Sum_{m=0..n} S(n, m) * T(2*m, x).

Links

Table of n, a(n) for n=0..65.

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n, m) = numerator(R(n, m)), n >= 0, m = 1, ..., n, with the rationals R(n, m) given by R(n, 0) = (1/2^(2*n-1))*binomial(2*n,n)/2 and R(n ,m) = (1/2^(2*n-1))*binomial(2*n, n-m) for m =1..n, n >= 0.

Example

The triangle a(n, m) begins:
n\m     0     1    2    3    4   5  6   7 8 9
0:      1
1:      1     1
2:      3     1    1
3:      5    15    3    1
4:     35     7    7    1    1
5:     63   105   15   45    5   1
6:    231    99  495   55   33   3  1
7:    429  3003 1001 1001   91  91  7   1
8:   6435   715 1001  273  455  35 15   1 1
9:  12155 21879 1989 4641 1071 765 51 153 9 1
...
The rational triangle R(n, m) begins:
n\m  0       1      2     3      4     5  ...
0:   1
1:  1/2     1/2
2:  3/8     1/2    1/8
3:  5/16   15/32   3/16  1/32
4: 35/128   7/16   7/32  1/16  1/128
5: 63/256 105/256 15/64 45/512 5/256 1/512
...
row 6: 231/1024 99/256 495/2048 55/512 33/1024 3/512 1/2048,
row 7: 429/2048 3003/8192 1001/4096 1001/8192 91/2048 91/8192 7/4096 1/8192,
row 8: 6435/32768 715/2048 1001/4096 273/2048 455/8192 35/2048 15/4096 1/2048 1/32768,
row 9: 12155/65536 21879/65536 1989/8192 4641/32768 1071/16384 765/32768 51/8192 153/131072 9/65536 1/131072,
...
n=3: x^6 = (5/16)*T(0, x) + (15/32)*T(2, x)
  +(3/16)*T(4, x) + (1/32)*T(6,x).
  cos^6(x) = (5/16) + (15/32)*cos(2*x) +
    (3/16)*cos(4*x) + (1/32)*cos(6*x).
  sin^6(x) = (5/16) - (15/32)*cos(2*x) +
    (3/16)*cos(4*x) - (1/32)*cos(6*x).

Mathematica

T[MaxN_] := Function[{n}, With[{exp = Expand[(1/2)^(2 n) (Exp[I x] + Exp[-I x])^(2 n)]}, Prepend[ 2 Coefficient[exp, Exp[I 2 # x]] & /@ Range[1, n], exp /. {Exp[_] -> 0}]]][#] & /@ Range[0, MaxN];
T[5] // ColumnForm
T2[MaxN_] := Table[Inverse[Outer[Coefficient[#1, x, #2] &, Prepend[ChebyshevT[#, x] & /@ Range[2 MaxN], 1], Range[0, 2 MaxN]]][[n, m]], {n, 1, 2 MaxN, 2}, {m, 1, n, 2}]
T2[6] // ColumnForm (* Bradley Klee, Jun 14 2016 *)

Prog

(PARI) a(n, m) = if (m == 0, numerator((1/2^(2*n-1)) * binomial(2*n, n)/2), numerator((1/2^(2*n-1))*binomial(2*n, n-m)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n, k), ", ")); print()); \\ Michel Marcus, Jun 19 2016

Crossrefs

Cf. A127673, A127674, A244420, A244421, A273168.

Sequence in context: A195892 A195522 A273169 * A364113 A324284 A300791

Adjacent sequences: A273164 A273165 A273166 * A273168 A273169 A273170

Keyword

nonn,tabl,frac,easy

Author

Wolfdieter Lang, Jun 12 2016

Status

approved