OFFSET
0,2
COMMENTS
Generally, Sum_{k=0..n} x^k*(n!/k!)^2 is asymptotic to BesselI(0,2*sqrt(x))*(n!)^2
FORMULA
a(n) = (n^2+2)*a(n-1) - 2*(n-1)^2*a(n-2).
a(n) ~ 2*Pi*BesselI(0,2*sqrt(2)) * n^(2*n+1)/exp(2*n).
MATHEMATICA
Table[(n!)^2*Sum[2^j/(j!)^2, {j, 0, n}], {n, 0, 20}]
Total/@Table[2^k (n!/k!)^2, {n, 0, 20}, {k, 0, n}] (* Harvey P. Dale, Jun 10 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Aug 24 2013
STATUS
approved