A337154 a(n) = 4^n * (n!)^2 * Sum_{k=0..n} 1 / ((-4)^k * (k!)^2).

1, 3, 49, 1763, 112833, 11283299, 1624795057, 318459831171, 81525716779777, 26414332236647747, 10565732894659098801, 5113814721015003819683, 2945557279304642200137409, 1991196720809938127292888483, 1561098229114991491797624570673, 1404988406203492342617862113605699

Offset: 0

Author

Ilya Gutkovskiy, Jan 27 2021

Keywords

• nonn

Crossrefs

Cf. A001907, A073701, A336807, A337152, A337153, A337155.

Programs

Mathematica

Table[4^n n!^2 Sum[1/((-4)^k k!^2), {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 4 x), {x, 0, nmax}], x] Range[0, nmax]!^2

PARI

a(n) = 4^n * (n!)^2 * sum(k=0, n, 1 / ((-4)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 4*x).

a(0) = 1; a(n) = 4 * n^2 * a(n-1) + (-1)^n.