OFFSET
0,3
FORMULA
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 ).
a(n) = n! * Sum_{k=0..n} Stirling1(n,k) * Bell(k) / 2^(n-k).
D-finite with recurrence a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2.
a(n) ~ sqrt(Pi) * n^((3*n + 1)/2) / (2^(n/2) * exp((3*n + 1)/2 - sqrt(2*n))). - Vaclav Kotesovec, Jul 17 2021
MATHEMATICA
Table[(n!)^2 Sum[1/((n - 2 k)! 4^k k!), {k, 0, Floor[n/2]}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x + x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
PROG
(PARI) a(n) = (n!)^2 * sum(k=0, n\2, 1/((n-2*k)!*4^k*k!)); \\ Michel Marcus, Jul 17 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Jul 16 2021
STATUS
approved