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A082649
Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).
1
1, 4, 4, 16, 24, 9, 64, 128, 80, 16, 256, 640, 560, 200, 25, 1024, 3072, 3456, 1792, 420, 36, 4096, 14336, 19712, 13440, 4704, 784, 49, 16384, 65536, 106496, 90112, 42240, 10752, 1344, 64, 65536, 294912, 552960, 559104, 329472, 114048, 22176, 2160, 81, 262144, 1310720, 2785280, 3276800, 2329600
OFFSET
1,2
COMMENTS
Using arcsin(x) = Pi/2 - arccos(x), valid for -1 < = x <= 1, we find sin^2(k*arcsin(x)) = sin^2(k*arccos(x)) for k odd, while sin^2(k*arcsin(x)) = 1 - sin^2(k*arccos(x)) for k even. Thus the expansion of sin^2(n*x) in powers of cos(x) will produce a similar table of coefficients. See the example section below. - Peter Bala, Feb 02 2017
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011 (rows 0 to 140, flattened)
FORMULA
Coefficients are: 4^(n-1), (2n)4^(n-2), (2n)(2n-3)4^(n-3)/2!, (2n)(2n-4)(2n-5)4^(n-4)/3!, (2n)(2n-5)(2n-6)(2n-7)4^(n-5)/4!, (2n)(2n-6)(2n-7)(2n-8)(2n-9)4^(n-6)/5!...
G.f. as triangle: (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)). - Robert Israel, Dec 20 2017
EXAMPLE
sinh^2 x = sinh^2 x
sinh^2 2x = 4 sinh^4 x + 4 sinh^2 x
sinh^2 3x = 16 sinh^6 x + 24 sinh^4 x + 9 sinh^2 x
sinh^2 4x = 64 sinh^8 x + 128 sinh^6 x + 80 sinh^4 x + 16 sinh^2 x
sinh^2 5x = 256 sinh^10 x + 640 sinh^8 x + 560 sinh^6 x + 200 sinh^4 x + 25 sinh^2 x
From Peter Bala, Feb 02 2016: (Start)
sin^2(x) = 1 - cos^2(x);
sin^2(2*x) = -4*cos^4(x) + 4*cos^2(x);
sin^2(3*x) = 1 - (16*cos^6(x) - 24*cos^4(x) + 9*cos^2(x));
sin^2(4*x) = -64*cos^8(x) + 128*cos^6(x) - 80*cos^4(x) + 16*cos^2(x);
sin^2(5*x) = 1 - (256*cos^10(x) - 640*cos^8(x) + 560*cos^6(x) - 200*cos^4(x) + 25*cos^2(x)). (End)
MAPLE
g:= (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)):
S:= series(g, x, 15):
seq(seq(coeff(coeff(S, x, n), y, k), k=0..n), n=0..14); # Robert Israel, Dec 20 2017
MATHEMATICA
Table[Reverse[CoefficientList[1/x TrigExpand[Sinh[n ArcSinh[Sqrt[x]]]^2], x]], {n, 7}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
Abs[Table[CoefficientList[x^n Piecewise[{{1 - ChebyshevT[n, 1/Sqrt[x]]^2, Mod[n, 2] == 0}, {ChebyshevT[n, 1/Sqrt[x]]^2, Mod[n, 2] == 1}}], x], {n, 10}]] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
CROSSREFS
A001108 gives row sums.
Closely related to A123583 and A123588.
Sequence in context: A240035 A321677 A223819 * A156232 A053441 A065732
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 16 2003, suggested by Herb Conn
EXTENSIONS
More terms from Robert Israel, Dec 20 2017
STATUS
approved