Following description from https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

import torch
import matplotlib.pyplot as plt
plt.style.use('dark_background')

# log of bezel function approximation
# the sum must go to infinity, but we stop at j=100
def bezel(v,y,infinity=100):
    if not isinstance(y,torch.Tensor):
        y = torch.tensor(y)
    if not isinstance(v,torch.Tensor):
        v = torch.tensor(v)
    j = torch.arange(0,infinity)
    bottom = torch.lgamma(j+v+1)+torch.lgamma(j+1)
    top = 2*j*(0.5*y.unsqueeze(-1)).log()
    mult = (top-bottom)
    return (v*(y/2).log().unsqueeze_(-1)+mult)

def noncentral_chi2(x,mu,k):
    if not isinstance(mu,torch.Tensor):
        mu = torch.tensor(mu)
    if not isinstance(k,torch.Tensor):
        k = torch.tensor(k)
    if not isinstance(x,torch.Tensor):
        x = torch.tensor(x)
    
    # the key trick is to use log operations instead of * and / as much as possible
    bezel_out = bezel(0.5*k-1,(mu*x).sqrt())
    x=x.unsqueeze_(-1)
    return (torch.tensor(0.5).log()+(-0.5*(x+mu))+(x.log()-mu.log())*(0.25*k-0.5)+bezel_out).exp().sum(-1)

# count of normal random variables that we will sum
loc = torch.rand((5))
normal = torch.distributions.Normal(loc,1)

# distribution parameter, also named as lambda
mu = (loc**2).sum()

# count of simulated sums
events = 5000
Xs = normal.sample((events,))

# chi-square distribution
Y = (Xs**2).sum(-1)

t = torch.linspace(0.1,Y.max()+10,100)
dist = noncentral_chi2(t,mu,len(loc))

# plot produced hist againts computed density function
plt.title(f"k={len(loc)}, mu={mu:0.2f}")
plt.hist(Y,bins=int(events**0.5),density=True)
plt.plot(t,dist)