%I #7 Jan 27 2021 18:43:56

%S 1,6,121,5446,435681,54460126,9802822681,2401691556846,

%T 768541298190721,311259225767242006,155629612883621003001,

%U 94155915794590706815606,67792259372105308907236321,57284459169428986026614691246,56138769986040406306082397421081,63156116234295457094342697098716126

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

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

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

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

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

%Y Cf. A006040, A056546, A336804, A336805, A336807.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jan 27 2021