A Free Sample of Autocorrelation

R, 3 31 25 bytes

# changes the builtin to only return the acf
body(acf)=body(acf)[1:18]

Usage (returns an array with the autocorrelations)

(acf(c(2.4, 2.4, 2.4, 2.2, 2.1, 1.5, 2.3, 2.3, 2.5, 2),5))
# , , 1
#
#             [,1]
# [1,]  1.00000000
# [2,]  0.07659298
# [3,] -0.06007802
# [4,] -0.51144343
# [5,] -0.02912874
# [6,] -0.10468140

Background:

The 31 byte solution based on the original acf built in

function(n,h)c(acf(n,h,,F)$acf)

Note that the 3 byte option acf is the originalbuilt in that will plot (and print to 3 digits) while returning the required autocorrelation as an element in a list.

usage

 acf(c(2.4, 2.4, 2.4, 2.2, 2.1, 1.5, 2.3, 2.3, 2.5, 2),5)

output:

#    Autocorrelations of series ‘c(2.4, 2.4, 2.4, 2.2, 2.1, 1.5, 2.3, 2.3, 2.5, 2)’, by lag
#
#     0      1      2      3      4      5 
# 1.000  0.077 -0.060 -0.511 -0.029 -0.105 

If we want the correlations to display to more than 3 decimal places then 28 bytes will do it (or 31, if we want to suppress the plot)

# will still plot (28 bytes)
function(n,h)c(acf(n,h)$acf)
# won't plot (31 bytes)
function(n,h)c(acf(n,h,,F)$acf)

Jelly, 26 25 24 23 20 bytes

×L_SµḊ;0µƓС׹S€µF÷Ḣ

Try it online!

How it works

×L_SµḊ;0µƓС׹S€µF÷Ḣ  Main link. Input: x (list) STDIN: h (integer)

×L                    Multiply all items in x by the length of x.
  _S                  Subtract the sum of x from all products.
                      Let's call the result X.
    µ                 Begin a new monadic chain. Argument: t (list)
     Ḋ                Remove the first element of t.
      ;0              Append a 0 to the result.
        µ             Push the previous chain and begin a new one.
                      Argument: X
         Ɠ            Read h from STDIN.
          С          Repeat the Ḋ;0 chain h times, collecting the h+1 intermediate
                      results in a list A.
            ×¹        Multiply the vectors in A by X.
              S€      Compute the sum of each vectorized product.
                µ     Begin a new, monadic chain. Argument: S (sums)
                 F    Flatten S. This does nothing except creating a deep copy.
                   Ḣ  Pop the first element of S.
                  ÷   Divide all elements of the copy by the first element.

Python 3, 147 130 126 120 bytes

def p(h,x):x=[t-sum(x)/len(x)for t in x];return[sum(s*t for s,t in zip(x[n:],x))/sum(b*b for b in x)for n in range(h+1)]

This solution is probably going to be golfed further, It's just a start.

You can call it with:

p(5,[2.4, 2.4, 2.4, 2.2, 2.1, 1.5, 2.3, 2.3, 2.5, 2])