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E.g.f.: x/(1+3x-4x^3)=x/[1-T(3,x)], where T(3,x) is a Chebyshev polynomial.
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%I #4 Aug 20 2021 07:07:09

%S 0,1,-6,54,-552,6840,-97200,1577520,-28667520,578067840,-12798777600,

%T 308836281600,-8065907942400,226719600307200,-6824229456844800,

%U 219010610827008000,-7465397891567616000,269363867734241280000,-10256545055212904448000

%N E.g.f.: x/(1+3x-4x^3)=x/[1-T(3,x)], where T(3,x) is a Chebyshev polynomial.

%C "Bernoulli numbers" for x/[1-T(3,x)].

%F D-finite with recurrence a(n) +(n+4)*a(n-1) -2*n*(n-1)*a(n-2) -4*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Aug 20 2021

%p G:=x/(1+3*x-4*x^3): Gser:=series(G,x=0,23): 0,seq(n!*coeff(Gser,x^n),n=1..20); # yields the signed sequence

%t g[x_] = x/(-1 + ChebyshevT[3, x]) h[x_, n_] = Dt[g[x], {x, n}] a[x_] = Table[h[x, n], {n, 0, 50}]; b = a[0]

%K sign

%O 0,3

%A _Roger L. Bagula_, Jun 28 2005