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A337153
a(n) = 3^n * (n!)^2 * Sum_{k=0..n} 1 / ((-3)^k * (k!)^2).
3
1, 2, 25, 674, 32353, 2426474, 262059193, 38522701370, 7396358663041, 1797315155118962, 539194546535688601, 195727620392454962162, 84554332009540543653985, 42869046328837055632570394, 25206999241356188711951391673, 17014724487915427380567189379274, 13067308406719048228275601443282433
OFFSET
0,2
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 3*x).
a(0) = 1; a(n) = 3 * n^2 * a(n-1) + (-1)^n.
MATHEMATICA
Table[3^n n!^2 Sum[1/((-3)^k k!^2), {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 3 x), {x, 0, nmax}], x] Range[0, nmax]!^2
PROG
(PARI) a(n) = 3^n * (n!)^2 * sum(k=0, n, 1 / ((-3)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 27 2021
STATUS
approved