%I #12 Jan 28 2021 11:11:45
%S 1,1,9,161,5153,257649,18550729,1817971441,232700344449,
%T 37697455800737,7539491160147401,1824556860755671041,
%U 525472375897633259809,177609663053400041815441,69622987916932816391652873,31330344562619767376243792849,16041136416061320896636821938689
%N a(n) = 2^n * (n!)^2 * Sum_{k=0..n} 1 / ((-2)^k * (k!)^2).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 2*x).
%F a(0) = 1; a(n) = 2 * n^2 * a(n-1) + (-1)^n.
%t Table[2^n n!^2 Sum[1/((-2)^k k!^2), {k, 0, n}], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!^2
%o (PARI) a(n) = 2^n * (n!)^2 * sum(k=0, n, 1 / ((-2)^k * (k!)^2)); \\ _Michel Marcus_, Jan 28 2021
%Y Cf. A000354, A073701, A336804, A337153, A337154, A337155.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 27 2021