sqrt(a(n)) = upper left term in M^n as to the modulus of a polar term; M = an infinite square production matrix in which a column of (i, i, i, ...) is appended to the right of Pascal's triangle, as follows (with i = sqrt(-1)):
1, i, 0, 0, 0, ...
1, 1, i, 0, 0, ...
1, 2, 1, i, 0, ...
1, 3, 3, 1, i, ...
... (End)
a(n) = |B_n(i)|^2, where B_n(x) is the n-th Bell polynomial, i = sqrt(-1) is the imaginary unit. - Vladimir Reshetnikov, Oct 15 2017
a(n) ~ (n*exp(-1 + Re(LambertW(i*n)) / Abs(LambertW(i*n))^2) / Abs(LambertW(i*n)))^(2*n) / Abs(1 + LambertW(i*n)), where i is the imaginary unit. - Vaclav Kotesovec, Jul 28 2021
MAPLE
A121870a:= proc(a) local i, t:
i:=1: t:=0: for i to 100 do t:=t+evalf((i^(a-1))*(I)^i/(i)!): od:
