What is the expected value and expected variance of a log skew normal distribution?

In case I have the terminology wrong, I'm referring to data that is lognormal with some skew mild skew when it's log transformed so it would have params y ~ logskewnormal(loc, scale, alpha).

What is the expected value and expected variance of a log skew normal distribution?

In case I have the terminology wrong, I'm referring to data that is lognormal with some skew mild skew when it's log transformed so it would have params y ~ logskewnormal(loc, scale, alpha).

edit: Got a great answer from JimB, code below to reproduce the expected value of the mean!

import numpy as np
import scipy.stats as stats
from scipy.special import erf, erfc

a, mu, sig = 1.1, 3.9, 1.25
skewnorm_dist = stats.skewnorm(a=a, loc=mu, scale=sig)

exp_mu_hat = []
for _ in range(10000):
    X = skewnorm_dist.rvs(1000)
    Y = np.exp(X)
    exp_mu_hat.append( Y.mean() )
    
print( np.mean(exp_mu_hat) )

term1 = np.exp(mu + 0.5*(sig**2))
term2 = erf((a*sig) / (np.sqrt(2) * np.sqrt(a**2 + 1))) + 1
exp_mu = term1*term2
print(exp_mu)