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).
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
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 12 2016
STATUS
approved