A273167 - OEIS

%I #27 Dec 11 2019 07:37:09

%S 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,

%T 1,429,3003,1001,1001,91,91,7,1,6435,715,1001,273,455,35,15,1,1,12155,

%U 21879,1989,4641,1071,765,51,153,9,1,46189,20995,62985,4845,4845,969,4845,285,95,5,1

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

%C The denominator triangle is given in A273168.

%C 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).

%C   Compare this with A127673.

%C 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.

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

%C 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).

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F 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.

%e The triangle a(n, m) begins:

%e n\m     0     1    2    3    4   5  6   7 8 9

%e 0:      1

%e 1:      1     1

%e 2:      3     1    1

%e 3:      5    15    3    1

%e 4:     35     7    7    1    1

%e 5:     63   105   15   45    5   1

%e 6:    231    99  495   55   33   3  1

%e 7:    429  3003 1001 1001   91  91  7   1

%e 8:   6435   715 1001  273  455  35 15   1 1

%e 9:  12155 21879 1989 4641 1071 765 51 153 9 1

%e ...

%e The rational triangle R(n, m) begins:

%e n\m  0       1      2     3      4     5  ...

%e 0:   1

%e 1:  1/2     1/2

%e 2:  3/8     1/2    1/8

%e 3:  5/16   15/32   3/16  1/32

%e 4: 35/128   7/16   7/32  1/16  1/128

%e 5: 63/256 105/256 15/64 45/512 5/256 1/512

%e ...

%e row 6: 231/1024 99/256 495/2048 55/512 33/1024 3/512 1/2048,

%e row 7: 429/2048 3003/8192 1001/4096 1001/8192 91/2048 91/8192 7/4096 1/8192,

%e row 8: 6435/32768 715/2048 1001/4096 273/2048 455/8192 35/2048 15/4096 1/2048 1/32768,

%e row 9: 12155/65536 21879/65536 1989/8192 4641/32768 1071/16384 765/32768 51/8192 153/131072 9/65536 1/131072,

%e ...

%e n=3: x^6 = (5/16)*T(0, x) + (15/32)*T(2, x)

%e   +(3/16)*T(4, x) + (1/32)*T(6,x).

%e   cos^6(x) = (5/16) + (15/32)*cos(2*x) +

%e     (3/16)*cos(4*x) + (1/32)*cos(6*x).

%e   sin^6(x) = (5/16) - (15/32)*cos(2*x) +

%e     (3/16)*cos(4*x) - (1/32)*cos(6*x).

%t 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 T[5] // ColumnForm

%t 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}]

%t T2[6] // ColumnForm (* _Bradley Klee_, Jun 14 2016 *)

%o (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)));

%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(a(n,k), ", ")); print()); \\ _Michel Marcus_, Jun 19 2016

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

%K nonn,tabl,frac,easy

%O 0,4

%A _Wolfdieter Lang_, Jun 12 2016