A337155 - OEIS

%I #11 Jan 28 2021 11:12:36

%S 1,4,81,3644,291521,36440124,6559222321,1607009468644,514243029966081,

%T 208268427136262804,104134213568131402001,63001199208719498210604,

%U 45360863430278038711634881,38329929598584942711331474444,37563331006613243857104844955121,42258747382439899339242950574511124

%N a(n) = 5^n * (n!)^2 * Sum_{k=0..n} 1 / ((-5)^k * (k!)^2).

%F Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselJ(0,2*sqrt(x)) / (1 - 5*x).

%F a(0) = 1; a(n) = 5 * n^2 * a(n-1) + (-1)^n.

%t Table[5^n n!^2 Sum[1/((-5)^k k!^2), {k, 0, n}], {n, 0, 15}]

%t nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 5 x), {x, 0, nmax}], x] Range[0, nmax]!^2

%o (PARI) a(n) = 5^n * (n!)^2 * sum(k=0, n, 1 / ((-5)^k * (k!)^2)); \\ _Michel Marcus_, Jan 28 2021

%Y Cf. A001908, A073701, A336808, A337152, A337153, A337154.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jan 27 2021