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A082649 Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).

%I A082649 #25 Dec 20 2017 23:32:33
%S A082649 1,4,4,16,24,9,64,128,80,16,256,640,560,200,25,1024,3072,3456,1792,
%T A082649 420,36,4096,14336,19712,13440,4704,784,49,16384,65536,106496,90112,
%U A082649 42240,10752,1344,64,65536,294912,552960,559104,329472,114048,22176,2160,81,262144,1310720,2785280,3276800,2329600
%N A082649 Triangle of coefficients in expansion of sinh^2(n*x) in powers of sinh(x).
%C A082649 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
%H A082649 Robert Israel, <a href="/A082649/b082649.txt">Table of n, a(n) for n = 1..10011</a> (rows 0 to 140, flattened)
%F A082649 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!...
%F A082649 G.f. as triangle: (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)). - _Robert Israel_, Dec 20 2017
%e A082649 sinh^2 x = sinh^2 x
%e A082649 sinh^2 2x = 4 sinh^4 x + 4 sinh^2 x
%e A082649 sinh^2 3x = 16 sinh^6 x + 24 sinh^4 x + 9 sinh^2 x
%e A082649 sinh^2 4x = 64 sinh^8 x + 128 sinh^6 x + 80 sinh^4 x + 16 sinh^2 x
%e A082649 sinh^2 5x = 256 sinh^10 x + 640 sinh^8 x + 560 sinh^6 x + 200 sinh^4 x + 25 sinh^2 x
%e A082649 From _Peter Bala_, Feb 02 2016: (Start)
%e A082649 sin^2(x) = 1 - cos^2(x);
%e A082649 sin^2(2*x) = -4*cos^4(x) + 4*cos^2(x);
%e A082649 sin^2(3*x) = 1 - (16*cos^6(x) - 24*cos^4(x) + 9*cos^2(x));
%e A082649 sin^2(4*x) = -64*cos^8(x) + 128*cos^6(x) - 80*cos^4(x) + 16*cos^2(x);
%e A082649 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)
%p A082649 g:= (1+x*y)/((1-x*y)*(1-(4+2*y)*x+x^2*y^2)):
%p A082649 S:= series(g,x,15):
%p A082649 seq(seq(coeff(coeff(S,x,n),y,k),k=0..n),n=0..14); # _Robert Israel_, Dec 20 2017
%t A082649 Table[Reverse[CoefficientList[1/x TrigExpand[Sinh[n ArcSinh[Sqrt[x]]]^2], x]], {n, 7}] // Flatten (* _Eric W. Weisstein_, Apr 05 2017 *)
%t A082649 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 *)
%Y A082649 A001108 gives row sums.
%Y A082649 Closely related to A123583 and A123588.
%K A082649 nonn,tabl
%O A082649 1,2
%A A082649 _Gary W. Adamson_, May 16 2003, suggested by _Herb Conn_
%E A082649 More terms from _Robert Israel_, Dec 20 2017