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%I #12 Jan 28 2021 11:12:27
%S 1,3,49,1763,112833,11283299,1624795057,318459831171,81525716779777,
%T 26414332236647747,10565732894659098801,5113814721015003819683,
%U 2945557279304642200137409,1991196720809938127292888483,1561098229114991491797624570673,1404988406203492342617862113605699
%N a(n) = 4^n * (n!)^2 * Sum_{k=0..n} 1 / ((-4)^k * (k!)^2).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 4*x).
%F a(0) = 1; a(n) = 4 * n^2 * a(n-1) + (-1)^n.
%t Table[4^n n!^2 Sum[1/((-4)^k k!^2), {k, 0, n}], {n, 0, 15}]
%t nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 4 x), {x, 0, nmax}], x] Range[0, nmax]!^2
%o (PARI) a(n) = 4^n * (n!)^2 * sum(k=0, n, 1 / ((-4)^k * (k!)^2)); \\ _Michel Marcus_, Jan 28 2021
%Y Cf. A001907, A073701, A336807, A337152, A337153, A337155.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jan 27 2021
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