A337152 a(n) = 2^n * (n!)^2 * Sum_{k=0..n} 1 / ((-2)^k * (k!)^2).

1, 1, 9, 161, 5153, 257649, 18550729, 1817971441, 232700344449, 37697455800737, 7539491160147401, 1824556860755671041, 525472375897633259809, 177609663053400041815441, 69622987916932816391652873, 31330344562619767376243792849, 16041136416061320896636821938689

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

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

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

Mathematica

Table[2^n n!^2 Sum[1/((-2)^k k!^2), {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!^2

Prog

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

Crossrefs

Cf. A000354, A073701, A336804, A337153, A337154, A337155.

Sequence in context: A371369 A243682 A367160 * A159831 A133793 A319197

Adjacent sequences: A337149 A337150 A337151 * A337153 A337154 A337155

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 27 2021

Status

approved