This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A238390 #48 Apr 01 2018 07:36:33
%S A238390 1,1,4,35,546,13482,485892,24108513,1576676530,131451399794,
%T A238390 13609184032808,1712978776719938,257612765775847132,
%U A238390 45620136452519144700,9396239458048330569840,2227147531572856811691105,601916577165056911293330930,183994483721828524163677628370
%N A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).
%C A238390 Aerated, the e.g.f. is e^(a.t) = 1/AC(i*t) = 1/[I_1(2i*t)/(it)] = 1/Sum_{n>=0} (-1)^n t^(2n) / [n!(n+1)!] = a_0 + a_2 t^2/2! + a_4 t^4/4! + ... = 1 + t^2/2! + 4 t^4/4! + 35 t^6/6! + ..., where AC(t) is the e.g.f. for the aerated Catalan numbers c_n of A126120 and I_n(t) are the modified Bessel functions of the first kind (i = sqrt(-1)). The signed, aerated sequence b_n = (i)^n a_n has the e.g.f. e^(b.t) = 1/AC(t) and, therefore, (i*a. + c.)^n = Sum_{k=0..n} binomial(n,k) i^k a_k c_(n-k) vanishes except for n=0 for which it's unity. - _Tom Copeland_, Jan 23 2016
%C A238390 With q(n) = A126120(n+1) and q(0) = 0, d(2n) = (-1)^n A238390(n) and zero for odd arguments, and r(2n+1) = (-1)^n A180874(n+1) and zero for even arguments, then r(n) = (q. + d.)^n = Sum_{k=0..n} binomial(n,k) q(k) d(n-k), relating these sequences (and A000108) through binomial convolutions. Then also, (r. + c. + d.)^n = r(n). See A180874 for proofs and for relations to A097610. For quick reference, q = (0, 1, 0, 2, 0, 5, 0, 14, ..), d = (1, 0, -1, 0, 4, 0, -35, 0, ..), and r = (0, 1, 0, -1, 0, 5, 0, -56, ..). - _Tom Copeland_, Jan 28 2016
%C A238390 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
%H A238390 G. C. Greubel, <a href="/A238390/b238390.txt">Table of n, a(n) for n = 0..98</a>
%F A238390 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
%p A238390 S:= series(x/BesselJ(1,2*x),x,102):
%p A238390 seq((2*j)!*coeff(S,x,2*j),j=0..50); # _Robert Israel_, Jan 31 2016
%t A238390 Table[(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n, 1, 41, 2}]
%Y A238390 Cf. A103365, A180874, A115369, A000275, A002190.
%Y A238390 Cf. A000108, A007318, A097610, A126120, A133314.
%K A238390 nonn
%O A238390 0,3
%A A238390 _Vaclav Kotesovec_, Mar 01 2014