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a(n) ~ c * (n!)^2 / (sqrt(n) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.91762316513274332857623611, and c = 1/(Sqrt[Pi]*BesselJ[2, BesselJZero[1, 1]]) = 1.4008104828035425937394082168... - _Vaclav Kotesovec_, Mar 01 2014, updated Apr 01 2018
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seq((2*j)!*coeff(S, x, 2*j*(2*j)!), j=0..50); # Robert Israel, Jan 31 2016
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S:= series(x/BesselJ(1, 2*x), x, 102):
seq(coeff(S, x, 2*j*(2*j)!), j=0..50); # Robert Israel, Jan 31 2016
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Aerated and signed, this sequence contains the moments m(n) of the Appell polynomial sequence UMT(n,h1,h2) that is the umbral compositional inverse of the Appell sequence of Motzkin polynomials MT(n,h1,h2) of A097610 with exp[x UMT(.,h1,h2)] = e^(x*h1) / AC(x*y) where y = sqrt(h2) and AC is defined above. UMT(n,h1,h2) = (m.y + h1)^n with (m.)^(2n) = m(2n) = (-1)^n A238390(n) and zero otherwise. Consequently, the associated lower triangular matrices A007318(n,k)*m(n-k) and A007318(n,k)*A126120(n-k) form an inverse pair, (cf. also A133314), and MT(n,UMT(.,h1,h2),h2) = h1^n = UMT(n,MT(.,h1,h2),h2). - Tom Copeland, Jan 30 2016
Aerated and signed, this sequence contains the moments m(n) of the Appell polynomial sequence UMT(n,h1,h2) that is the umbral compositional inverse of the Appell sequence of Motzkin polynomials MT(n,h1,h2) of A097610 with exp[x UMT(.,h1,h2)] = e^(x*h1) / AC(x*y) where y = sqrt(h2) and AC is defined above. UMT(n,h1,h2) = (m.y + h1)^n with (m.)^(2n) = m(2n) = (-1)^n A238390(n) and zero otherwise. Consequently, the associated lower triangular matrices T(n,k) = A007318(n,k)*m(n-k) and TA007318(n,k)*A126120(n-k) form an inverse pair, and MT(n,UMT(.,h1,h2),h2) = h1^n = UMT(n,MT(.,h1,h2),h2). - Tom Copeland, Jan 30 2016